Welcome! - Mathematics at The Sultan's School

Level 9/10 Pack 1 Page 1. [email protected]

Welcome!

This is the first in a series of teaching aids designed by teachers for teachers at level 9/10.Although level 9 and level 10 notations have now ceased to be used, we are using them forcontinuity in our series. At Key Stage 3 these levels of work would be the Extension paper and inGCSE would represent work at the Higher level. The worksheets are designed to support thedelivery of the National Curriculum in a variety of teaching and learning styles. They are notdesigned to take the pedagogy away from the teacher. The worksheets are centred around theshown level, but spiral from the level below to the level above. Consult the National NumeracyStrategy for definitive National Curriculum levels. They can be used by parents with the supportof the on-line help facility at www.10ticks.co.uk.

Contents and Teacher Notes.

Pages 3/4. Accuracy of Measurement Notes.Notes about accuracy of measurement and the lower and upper bound.This includes four rules calculations with rounded numbers and finding the rangein which the answers are valid.

Pages 5/6. Accuracy of Measurement 1.Four rules calculations with rounded numbers, calculating the answer and then thelower and upper bounds of the answers.

Pages 7/8. Accuracy of Measurement 2.Calculating the percentage error and the maximum percentage error for four rulescalculations with rounded numbers.

Pages 9/10. Accuracy of Measurement 3.Typical examination questions covering accuracy of measurement.

Pages 11/12. Variation 1.The worksheets revise direct proportion and concepts which pupils should alreadybe familiar with. This is then extended to direct proportion including squares,cubes and square roots.

Pages 13/14. Variation 2.The first page deals with basic inverse proportion. This is then extended toindirect proportion including squares, cubes and square roots.

Pages 15/16. Direct and Indirect Proportion.Typical examination questions covering direct and indirect proportion.

Pages 17/18. Indices (Powers).Introducing negative indices by pattern and then evaluating numbers involvingnegative indices. Indices that represent roots. Whole numbers and fractions thathave negative and fraction indices.

Pages 19/20. Indices 1/2.Patterns using indices.


Pages 21/22. Compound Growth and Decay 1. (Exponential Functions).Questions that show compound growth and graphs. Investigation to highlight thedifference between uniform and compound growth. Compound decay.

Pages 23/24. Compound Growth and Decay 2. (Exponential Functions).Typical examination questions covering exponential functions.

Pages 25/26. Exponential Functions.Carbon dating and radioactivity. Notes on radiometric dating.

Pages 27/28. Surds.Multiplying and dividing surds. Rationalising expressions that contain surds.

Pages 29/30. Rational and Irrational Numbers 1.Showing numbers to be rational. Converting decimals and recurring decimal tosimple fractions.

Pages 31/32. Rational and Irrational Numbers 2.Irrational numbers. Which are rational numbers and which are irrational numbers?Typical examination questions covering rational and irrational numbers.

Page 33. Using Surds in Trigonometry.Using the trigonometrical ratios and Pythagoras’ Theorem to find lengths of sidesof right angled triangles.

Page 34. Trigonometric Ratios for 45˚, 30˚ and 60˚ in Surd Form.Finding these values.

Page 35. Using Trigonometric Ratios for 45˚, 30˚ and 60˚ in Surd Form.Finding missing values in diagrams using the trigonometric ratios for 45˚, 30˚ and60˚ in surd form.

Pages 36/37. Investigations with Irrational and Prime Numbers.Seven investigations, four of which are enjoyable paper folding problems.

Page 38. Proving Irrationality.Proof by contradiction.

Pages 39/40. Calculating π.Using spreadsheets to calculate π. Pierre de Fermat’s Prime Numbers and more onrationalising.

Page 41. Approximating √2 by Iteration.As it says.

Page 42. Powers of 10/Logarithms.Investigating decimal powers of 10 in a spreadsheet. Logarithms.

Copyright in Worksheets. ©Fisher Educational Ltd. 2002.Copyright in the worksheets belongs to Fisher Educational Ltd. Each purchase of the worksheets represents alicence to use and reproduce the worksheets as set out in the Terms and conditions shown on the 10ticks website.

'10TICKS', and '10TICKS.co.uk' and/or other 10TICKS services referenced on this web site or on the Worksheetsare trademarks of Fisher Educational Ltd. in the UK and/or other countries.

Details of copyright ownership in the clip art used in these worksheets:Copyright in the clip art used entirely in this pack is owned by Nova Development Corporation, California, USA.


Accuracy of Measurement Notes.

We have dealt with questions, such as "Round 4.345 to 2 decimal places" and" Round 45683 to 2significant figures". In this section we are considering these questions in reverse !

When a measurement is given as 4 m, this implies it is measured to the nearest metre. When ameasurement is given as 4.0 m, this implies it is measured to the nearest 0.1 of a metre, and whena measurement is given as 4.00 m, this implies it is measured to the nearest 0.01 of a metre. Allthese measurements are not precise. Let us consider the 4 m measurement. If this is to thenearest metre it is possible that the measurement could have been from 3.5 m (the lower bound)all the way to 4.5 m (the upper bound).

The lower bound fits into our understanding well, if you gave 3.5 to 1 significant figure it wouldbe 4. Notice the upper bound. In our example the upper bound 4.5 to 1 significant figure would be5. To be more precise the upper bound should really be < 4.5 i.e. 4.49 or 4.499 or 4.4999 etc.This when rounded to 1 significant figure would give 4. These numbers are getting closer andcloser to 4.5 so we say that 4.5 is the upper bound.

Similarly, 15 to the nearest whole number can be represented by any number from 14.5 to 15.5.

For any value given to the nearest unit

i). the maximum possible error is half of that "unit". ii). its limits of accuracy are the value ± half of the "unit".

For example a). An answer is given as 42 to 2 significant figures. So the maximum error is 0.5 (half of 1). Its limits of accuracy are 42 ± 0.5 . Or lower bound 41.5, upper bound 42.5. b). An answer is given as 480 to 2 significant figures. So the maximum error is 5 (half of 10). Its limits of accuracy are 480 ± 5 . Or lower bound 475, upper bound 485. c). An answer is given as 73000 to 2 significant figures. So the maximum error is 500 (half of 1000). Its limits of accuracy are 73000 ± 500 . Or lower bound 72500, upper bound 73500. d). An answer is given as 8.7 to 1 decimal place. So the maximum error is 0.05 (half of 0.1). Its limits of accuracy are 8.7 ± 0.05 . Or lower bound 8.65, upper bound 8.75. e). An answer is given as 6.99 to 2 decimal places. So the maximum error is 0.005 (half of 0.01). Its limits of accuracy are 6.99 ± 0.005 . Or lower bound 6.985, upper bound 6.995.

3 3.5 4 4.5 5

Any numbers in this range are given as 4


A value used in a calculation may not be an exact number, it may be an approximation whoselevel of accuracy and error is known. The approximation may be the result of rounding a numberor measuring to a certain degree of accuracy. This level of accuracy will have a bearing on thefinal result and is typical of what a GCSE question will want to find out.

Note the following as these tend to be the crux of GCSE questions :

When adding, the lower bound of the result is the sum of the lower limits of the two values. Theupper bound is the sum of the upper limits of the two values. To assist us with the upper andlower bounds using subtraction this diagram will help:

smallernumber

bigger number

Least difference

Greatest difference

For the biggest possible number For the smallest possible number

When multiplying biggest no. x biggest no. smallest no. x smallest no.When dividing biggest no. ÷ smallest no. smallest no. ÷ biggest no.

Examples: For these examples the numbers have been given to 1 decimal place.

Adding. usual answer lower bound upper bound 3.7 3.65 3.75 4.6 + 4.55 + 4.65 + 8.3 8.20 8.40

Therefore the answers accuracy lies between 8.20 and 8.40.

Subtracting. usual answer lower bound upper bound 7.7 7.65 7.75 4.6 - 4.65 - 4.55 - 3.1 3.00 3.20


Multiplying. usual answer lower bound upper bound 3.7 3.65 3.75 4.6 x 4.55 x 4.65 x 17.02 16.6075 17.4375


Dividing. usual answer lower bound upper bound 5.5 = 5 5.45 = 4.739 5.55 = 5.286 1.1 1.15 1.05



Accuracy of Measurement 1.

1). Give the lower and upper bounds of these numbers.

a). 4.43 b). 0.54 c). 3.264 d). 46 e). 3.04f). 150 (given to the nearest 10) g). 6800 (given to the nearest 100)h). 21.0 i). 65 j). 0.07 k). 3 l). 3.0m). 3.00 n). 45.00 o). 79 p). 3400 (given to 2 sig.fig.)q). 3400 (given to 3 sig.fig.) r). 4.98 s). 4567

2). In the following questions the numbers given are all to the nearest whole number.Work out the answer and then the lower and upper bounds of the answers.

a). 65 + 78 b). 4 + 3 c). 7 + 32 d). 97 + 88e). 98 - 46 f). 92 - 79 g). 9 - 2 h). 45 - 9i). 3 x 60 j). 92 x 3 k). 50 x 21 l). 49 x 12m). 48 ÷ 5 n). 54 ÷ 9 o). 69 ÷ 20 p). 312 ÷ 21

3). In the following questions the numbers given are all to 2 significant figures.Work out the answer and then the lower and upper bounds of the answers.

a). 65 + 470 b). 45 + 370 c). 780 + 320 d). 490 + 88e). 7.9 - 4.6 f). 24 - 7.9 g). 980 - 29 h). 9.0 - 6.7i). 3.0 x 6.4 j). 2.9 x 3.0 k). 5.0 x 2.9 l). 4.6 x 6.4m). 78 ÷ 5.0 n). 52 ÷ 9.4 o). 6900 ÷ 290 p). 3900 ÷ 92

4). In the following questions the numbers given are all to 1 decimal place.Work out the answer and then the lower and upper bounds of the answers.

a). 2.5 + 4.7 b). 0.4 + 0.3 c). 0.8 + 3.2 d). 4.7 + 8.8e). 7.8 - 4.6 f). 14.2 - 7.9 g). 3.1 - 2.3 h). 9.5 - 6.0i). 3.0 x 6.8 j). 2.2 x 3.1 k). 5.0 x 2.1 l). 4.7 x 1.5m). 4.8 ÷ 5.0 n). 5.8 ÷ 9.2 o). 6.9 ÷ 2.9 p). 3.7 ÷ 9.1

5). In the following questions the numbers given are all to 2 decimal places.Work out the answer and then the lower and upper bounds of the answers.

a). 2.65 + 4.78 b). 0.04 + 0.37 c). 0.78 + 3.26 d). 4.97 + 8.83e). 7.98 - 4.61 f). 8.92 - 7.90 g). 3.91 - 2.03 h). 9.45 - 6.09i). 3.02 x 6.08 j). 2.92 x 3.21 k). 5.00 x 2.91 l). 4.97 x 1.25m). 4.78 ÷ 5.90 n). 5.28 ÷ 9.42 o). 6.92 ÷ 2.90 p). 3.71 ÷ 9.21

6). In the following questions the first number is given to 3 significant figures and the secondnumber is given to the nearest whole number.Work out the answer and then the lower and upper bounds of the answers.

a). 265 + 4 b). 0.04 + 3 c). 4780 + 32 d). 497 + 83e). 798 - 461 f). 4920 - 79 g). 291 - 23 h). 90.4 - 6i). 3.02 x 6 j). 292 x 2 k). 57800 x 2 l). 497 x 25m). 478 ÷ 5 n). 5.28 ÷ 9 o). 69200 ÷ 290 p). 3710 ÷ 92


7). Two sides of a rectangle were measured to the nearest mm, as 4.6 cm and 7.9 cm.a). Find the least and greatest possible values of the perimeter.b). Find the least and greatest possible values of the area.

8). Two sides of a rectangle were measured to the nearest m, as 12 m and 26 m.a). Find the least and greatest possible values of the perimeter.b). Find the least and greatest possible values of the area.

9). Two sides of a rectangle were measured to the nearest cm, as 4.51 m and 7.90 m.a). Find the least and greatest possible values of the perimeter.b). Find the least and greatest possible values of the area.

10). A rectangular field was measured to the nearest 10 m, as 1.23 km wide and 2.48 km long.a). Find the least and greatest possible values of the perimeter.b). Find the least and greatest possible values of the area.

11). A wooden rod was measured as 132.4 cm. After a length was cut off the new length was63.2 cm. The measurements were made to the nearest mm.What are the greatest and least values of the reduction in length ?

12). Two rods are measured and found to be 10.3 cm and 4.7 cm to the nearest mm.What are the greatest and least values of the lengths ofa). each rod,b). the two rods placed end to end ?

13). Bill measured the dimensions of his rectangular shaped garden. He found the lengthwas 23.7 m and the width was 12.3 m, each measurement being correct to the nearesttenth of a metre.a). Between what limits must the true length and width lie ?b). Calculate the upper and lower limits for the area of his garden.

14). The weight of 1 cm3 of silver is stated to be 10.6 g to 3 significant figures.What are the lower and upper bounds for the weight of 30 cm3 of silver ?

15). The weight of 1 cm3 of gold is stated to be 19 g to 2 significant figures.What are the lower and upper bounds for the weight of 80 cm3 of gold ?

16). The weight of 1 cm3 of petrol is stated to be 0.8 g to 1 decimal place.What are the lower and upper bounds for the weight of 50 cm3 of petrol ?

17). Four packages are weighed, each to the nearest gram.Their weights are 453 g, 264 g, 836 g and 261 g.What are the lower and upper bounds for the total weight ?

18). A square playground has a side length 45 m, to the nearest metre.a). What are the lower and upper bounds for the perimeter ?b). What are the lower and upper bounds for the area?



The Percentage Error.

The error is the difference between the estimated value and the actual true value. Therefore

Percentage error = error x 100 true value

E.g. If you measure a line as 6.4 cm, then the upper bound is 6.45 and the lower bound is 6.35.As we do not know the actual true length we will use the measured length as an estimate.The maximum error possible = 0.05.

The Maximum percentage error = 0.05 x 100 = 0.78 % ( 2 d.p.) 6.4

1). Find the percentage error for these measurements.a). 470 metres, measured to the nearest ten metres.b). 3.2 cm, measured to the nearest mm.c). 4600 mm measured to 2 significant figures.d). 3.45 m measured to 2 decimal places.e). 0.0345 measured to 3 significant figures.f). 350 cm measured to 2 significant figures.g). 23.4 mm measured to 1 decimal place.h). 15.3 m measured to 1 decimal place.i). 34000 m measured to the nearest thousand metres.j). 0.3 m measured to 1 decimal place.

2). Find the percentage error below.a). A measured length was 6.2 cm, the true length was calculated at 6.22 cm.b). A measured weight was 24.3 kg, the true weight was calculated at 24.24 kg.c). A measured volume was 2.0 m3, the true volume was calculated at 2.04 m3.d). A measured length was 43.2 m, the true length was calculated at 43.18 m.e). π was estimated at 3.14. A closer approximation is 3.14159.f). √2 was estimated at 1.41. Use the value given on your calculator.

3). Two rods are measured and found to be 4.6 cm and 5.8 cm, to the nearest mm.The rods are placed end to end.a). What is the greatest length possible of the new rod ?b). What is the least possible length of the new rod ?c). What percentage error is possible ?

4). Four identical blocks are weighed to the nearest 100g. Each weighs 1.4 kg.The blocks are placed end to end.a). What is the greatest possible weight of the 4 blocks together ?b). What is the least possible weight of the 4 blocks together ?c). What percentage error is possible ?


The Maximum Percentage Error.

The only estimate in the previous calculation was the ‘true value’.

Percentage error = error x 100 true value

To maximise this calculation we need to make the ‘true value’ as small as possible.So if we repeat the example on the last page.

E.g. If you measure a line as 6.4 cm, then the upper bound is 6.45 and the lower bound is 6.35.As we do not know the actual true length we will use the lower bound.The maximum error possible = 0.05.

The Maximum percentage error = 0.05 x 100 = 0.79 % ( 2 d.p.)6.35

Hence Maximum Percentage Error = largest possible error x 100 least possible value

1). Find the maximum percentage error for these measurements.a). 470 metres, measured to the nearest ten metres.b). 3.2 cm, measured to the nearest mm.c). 4600 mm measured to 2 significant figures.d). 3.45 m measured to 2 decimal places.e). 0.0345 measured to 3 significant figures.f). 350 cm measured to 2 significant figures.g). 23.4 mm measured to 1 decimal place.h). 15.3 m measured to 1 decimal place.i). 34000 m measured to the nearest thousand metres.j). 0.3 m measured to 1 decimal place.

Compare these with your answers on the previous page.

2). The area of a rectangle is measured at 4.6 cm by 5.8 cm to the nearest mm.a). Find the greatest possible area of the rectangle.b). Find the smallest possible area of the rectangle.c). Find the maximum percentage error possible.

3). The area of a rectangular playground is measured at 35.6 m by 18.7 m to 1 decimal place.a). Find the greatest possible area of the playground.b). Find the smallest possible area of the playground.c). Find the maximum percentage error possible.

4). The length of one side of a large square is 3.4 m to the nearest 10 cm.Find the maximum possible percentage error of the area of the square.

5). The length of one side of a large square field is 2.3 km to 1 decimal place.Find the maximum possible percentage error of the area of the field.



1). Petula measures the width of her page as "14 cm exactly". Her ruler is marked in cmand mm. Comment on Petula's use of the word "exact".

2). The sides of a rectangle are measured as 5.00 m and 6.00 m.a). Calculate the greatest possible value of the perimeter.b). Calculate the least possible area.

3). Bob and Bill were asked their age. Both said 12. What is the greatest possible age gap ?

4). Kate read the thickness of a piece of wood as 14.80 mm. She told her friend thismeant exactly the same as 14.8 mm. Explain why she was wrong.

5). The sides of a rectangular garden are measured as 3.2 m and 7.4 m.a). Calculate the greatest possible value of the perimeter.b). Calculate the least possible area.

6). William read the thickness of a tree as 1.4 m.He told his friend that this meant exactly the same as 140 cm. Explain why he was wrong.

7). The measurements of a stamp are given as length 10 mm and width 6 mm, correct to thenearest mm.a). Between what limits must the length of the stamp lie ?b). Between what limits must the area of the stamp lie ?c). The area is given as (60± x) mm2. Suggest a suitable value for x.

8). A table top measures 55 cm by 135 cm correct to the nearest cm.a). What are the maximum side lengths of the table top ?b). What is the minimum area of the table top ?c). The perimeter of the table top is given as 3.80 m.

What is the greatest percentage error in the perimeter ?

9). Circular holes are drilled in a piece of metal.The holes have a radius of 1.4 cm, correct to 1 d.p..Bolts are made to fit these holes. The bolts have a radius of 1.30 cm correct to 2 d.p..Clearance is defined as the difference between the two radii.a). What is the maximum clearance between the bolt and the hole ?b). What is the minimum clearance between the bolt and the hole ?

10). The measurements of a label on a box are given as length 18 mm and width 9 mm,correct to the nearest mm.a). Between what limits must the length of the label lie ?b). Between what limits must the area of the label lie ?c). The area is given as (162± x) mm2. Suggest a suitable value for x.

11). A rectangular table top measures 58 cm by 192 cm correct to the nearest cm.a). What are the maximum side lengths of the table tops ?b). What is the minimum area of the table top ?c). The perimeter of the table top is given as 5.00 m.

What is the greatest percentage error in the perimeter ?


12). A petrol pump shows the volume bought to the nearest 0.01 litre.Bertie was supplied with 24.36 litres of petrol.a). What are the minimum and maximum amounts he could have obtained ?b). In one day 894 people use this petrol pump.

During the day the volumes of petrol shown on the pump are added to give a total.Each time the pump is used the minimum volume of petrol is given.What is the difference between the volume of petrol supplied and the totalof the pump readings?

13). A stop watch records the winner of 100 m race as 11.6 seconds, measured to the nearestone tenth of a second.a). What are the greatest and least possible times for the winner ?b). The length of the track is correct to the nearest 10 cm.

What are the greatest and least lengths of the track ?c). What is the fastest possible average speed of the winner clocking 11.6 s

in the100 m race ?

14). The volume of a cylinder is given as 840 ml correct to 2 significant figures.Peter measures its height as 16.8 cm (to the nearest mm).Between what limits must the radius of the cylinder lie ?

15). The weight of a jar of jam is 320 g. During a sales promotion it is increased to 380 g.Both weights are given to correct to 2 significant figures."18.5 % extra, 380 g for the price of 320 g" the adverts say.Within what limits does the percentage increase actually lie ?

16). The volume of a cylinder is given as 1200 ml correct to 2 significant figures.John measures its height as 12.1 cm (to the nearest mm).Between what limits must the radius of the cylinder lie ?

17). The volume of a can of cola is 420 ml. During a sales promotion it is increased to 500 ml.Both volumes are given to correct to 2 significant figures."19 % extra, 500 ml for the price of 420 ml" the adverts say.Within what limits does the percentage increase actually lie ?

18). A stop watch records the winner of 400 m race as 56.1 seconds, measured to the nearestone tenth of a second.a). What are the greatest and least possible times for the winner ?b). The length of the track is correct to the nearest 1 m.

What are the greatest and least possible lengths of the track ?c). What is the fastest possible average speed of the winner clocking 56.1 s in

the 400 m race ?

19). The volume of a can of beer is 500 ml. During a sales promotion it is increased to 620 ml.Both volumes are given to correct to 2 significant figures."24 % extra, 620 ml for the price of 500 ml" the adverts say.Within what limits does the percentage increase actually lie ?

20). The volume of a small cylindrical drum is given as 3400 ml correct to 2 significant figures.John measures its height as 15.6 cm (to the nearest mm).Between what limits must the radius of the cylinder lie ?


Variation 1.Direct Proportion.Revision.

The term ‘direct proportion’ has the same meaning as ‘direct variation’.

1). y is directly proportional to x. y = 15 when x = 2.5.a). Write an equation in y and x.b). Find k, the constant of proportionality.c). Hence, write a new equation in y and x.

2). P is directly proportional to V. P = 13.3 when V = 1.4.a). Write an equation in P and V.b). Find k, the constant of proportionality.c). Hence, write a new equation in P and V.

3). A α l and A = 36 when l= 9. Calculate A when l =12.4). If w is directly proportional to d and w = 24 when d = 6, find w when d = 7.5). r is directly proportional to s. If r = 21 when s = 3, find

a). r when s = 9, b). s when r = 70.6). f is directly proportional to g. If f = 8 when g = 12, find

a). f when g = 18, b). g when f = 16.7). C is directly proportional to D. If C = 39.9 when D = 7, find

a). C when D = 18, b). D when C = 125.4.8). If y α x and y = 4 when x = 2, find y when x = 5, and x when y = 6.9). If x α y and x = 15 when y = 1.5, find x when y = 6, and y when x = 3.5.10). y α x and y = 3 when x = 1.

Find the value of y when x = 6 and the value of x when y = 7.5.11). If y varies directly as x and y = 1/2 when x = 5, find y when x = 20.12). y varies directly as x and y = 2 when x = 3.

Calculate y when x = 9 and the value of x when y = 0.5.13). If the cost of 15 pears is €1.65.

a). Write a formula linking the cost and the number of pears.b). Find the cost of 19 pears.c). How many pears would you get for €3.41 ?

14). The distance covered by a car is directly proportional to the time taken.The car covers 171 km in 4.5 hours.a). Find how far it covers in 2.5 hours.b). If it travels 123.5 km, how long has it been travelling ?

15). The number of exercise books in a pile is directly proportional to its height. A pile of20 exercise books is 18 cm high. How high would a pile of 52 such books be ?

16). The length of dress material is directly proportional to its cost.Three metres of dress material cost €10.35, what is the cost of 8.4 metres ?

Squares, Cubes and Square Roots.

These questions follow the same format as before.

E.g. The weight, in g, of a ball bearing varies directly with the cube of the radius,measured in mm. A ball bearing of radius 3 mm has a weight of 43.2 g.a). What will a ball bearing of radius 7 mm weigh ?b). A ball bearing weighs 200 g. What is its radius ?


w α r3 w = kr3 43.2 = k33

43.2 = 27k k = 1.6 so w = 1.6r3

a). When r = 7 mm w = 1.6 x 73 w = 548.8gb). When w = 200g 200 = 1.6 r3 125 = r3 r = 5 mm.

1). y is directly proportional to the square of x. y = 100 when x = 5.a). Write an equation in y and x.b). Find k, the constant of proportionality.c). Hence, write a new equation in y and x.

2). P is directly proportional to the square root of V. P = 4.8 when V = 16.a). Write an equation in P and V.b). Find k, the constant of proportionality.c). Hence, write a new equation in P and V.

3). T is directly proportional to the cube of S. T = 68 and S = 2.a). Write an equation in T and S.b). Find k, the constant of proportionality.c). Hence, write a new equation in T and S.

4). If A is proportional to r2 and A = 20 when r = 2, find the value of A when r = 5.5). If y α x2 and y = 27 when x = 3. Find y when x = 7, and x when y = 12.6). If y varies as the square of x and y = 4 when x = 4, find the value of y when x = 5.7). V α r2 and V = 96 when r = 4. Find r when V =150 and r when V =121.5.8). y α x3 and y = 3 when x =1. Find the value of y when x = 2.9). If y varies as the cube of x and y =1000 when x = 5, find the value of y when x =10.10). y α √x and y =1 when x = 4. Calculate the value of y when x = 6.25.11). If p varies as the square root of q and p =7 when q = 25, find the value of p when q = 64.12). y varies as the square root of x and y = 2 when x = 2.25.

Find the value of y when x = 9 and the value of y when x = 0.25.13). r is directly proportional to s3. If r = 300 when s = 10, find

a). r when s = 6, b). s when r = 153.6.14). f is directly proportional to √g. If f = 5.4 when g = 9, find

a). f when g = 81, b). g when f = 7.2.15). C is directly proportional to D2. If C = 270 when D = 6, find

a). C when D = 4.4, b). D when C = 1080.16). r is directly proportional to √s. If r = 6 when s = 0.25, find

a). r when s = 12.25, b). s when r = 8.17). f is directly proportional to g2. If f = 5 when g = 5, find

a). f when g = 10, b). g when f = 2.888.18). C is directly proportional to D3. If C = 9 when D = 3, find

a). C when D = 2, b). D when C = 72.19). y α x2 and y = 3 when x = 2. Calculate the value of y when x = 1, and x when y = 3.20). y varies as the square of x and y = 9 when x = 6.

Calculate the value of y when x = 4 and the value of x when y = 16.21). V α r3 and V = 243 when r = 3. Calculate V when r = 4.5. When V = 1944 calculate r.22). If y α √z and y = 1.5 when z = 1/

4, find y when z = 16, and z when y = 9.

23). T α √ l and T = 6 when l = 6.25. Calculate T when l =1.96 and l when T = 4.24). The weight of a metal sphere varies directly with the cube of its radius.

The weight of a metal sphere of radius 3 cm is 4.06 kg.Calculate the weight of a metal sphere of radius 5 cm.

25). The weight of a metal disc varies directly with the square of its radius.If the weight of a metal disc of radius 5 cm is 400 g, what is the weight of asimilar disc of radius 9 cm ?


Variation 2.Inverse Proportion.

When one variable is directly proportional to the reciprocal of anothervariable the two are said to be inversely proportional to each other.

The product of the two variables is therefore constant.As one variable increases the other decreases.

E.g. P is inversely proportional to q and p = 6 when q = 2. Finda). p when q = 8, b). q when p = 1.5.

p α 1 p = k 6 = k k = 12 Therefore p = 12 q q 2 q

a). p = 12 p = 1.5 b). 1.5 = 12 q = 12 q = 8 8 q 1.5

1). y is inversely proportional to x. y = 6 when x = 1.5.a). Write an equation in y and x.b). Find k, the constant of proportionality.c). Hence, write a new equation in y and x.

2). P is inversely proportional to V. P = 4.3 when V = 9.a). Write an equation in P and V.b). Find k, the constant of proportionality.c). Hence, write a new equation in P and V.

3). T is inversely proportional to S. T = 21/2 when S = 4.

a). Write an equation in T and S.b). Find k, the constant of proportionality.c). Hence, write a new equation in T and S.

4). y α 1/x and y = 2 when x = 5. Calculate the value of y when x = 1.5). If y varies inversely as x and y = 8 when x = 3, find the value of y when x = 6.6). If y is inversely proportional to x and y =11/

2 when x = 3, what is the value of

y when x = 9?7). c varies inversely as b and c = 9 when b = 8. Calculate the values of

a). c when b = 4, b). c when b =12.8). r is inversely proportional to s. If r = 4.2 when s = 3, find

a). r when s = 2, b). s when r = 25.2.9). y varies inversely as x and y =1 when x = 6.

Calculate the values of y as x =1.5 and x as y = 9.10). If p α 1/q and p = 1/

4 when q = 8, find p when q = 6, and q when p = 3.

11). The amount of time for digging a hole is inversely proportional to the number of menworking. Four men dig a hole in 12 days.In how many days can six men dig the same size hole ?

12). The number of pumps pumping water is inversely proportional to the time taken topump a given amount of water. How long will it take 9 pumps working at the samerate to empty a tank, if 3 pumps can empty it in 12 hours ?

13). f is inversely proportional to (g+2). If f = 4 when g = 3, finda). f when g = 13, b). g when f = 5.

14). C is inversely proportional to (7 - D). If C = 2 when D = 5, finda). C when D = 3, b). D when C = 2/

3.

15). r is inversely proportional to (2s + 1). If r = 2 when s = 1, finda). r when s = 4, b). s when r = 3.


More Complex Inverse Proportion.

E.g. P is inversely proportional to the square root of q and p = 2 when q = 16. Finda). p when q = 25, b). q when p = 4.

p α 1 p = k 2 = k k = 8 Therefore p = 8 √q √q √16 √q

a). p = 8 p = 1.6 b). 4 = 8 √q = 8 √q = 2 q = 4 √25 √q 4

1). y is inversely proportional to the square of x. y = 3.5 when x = 2.a). Write an equation in y and x.b). Find k, the constant of proportionality.c). Hence, write a new equation in y and x.

2). P is inversely proportional to the square root of V. P = 5.1 when V = 9.a). Write an equation in P and V.b). Find k, the constant of proportionality.c). Hence, write a new equation in P and V.

3). T is inversely proportional to the cube of S. T = 21/4 when S = 2.

a). Write an equation in T and S.b). Find k, the constant of proportionality.c). Hence, write a new equation in T and S.

4). r is inversely proportional to s3. If r = 1/4 when s = 4, find

a). r when s = 1, b). s when r = 2.5). f is inversely proportional to √g. If f = 6 when g = 16, find

a). f when g = 4, b). g when f = 3.6). C is inversely proportional to D2. If C = 4 when D = 3, find

a). C when D = 2, b). D when C = 21/4.

7). r is inversely proportional to √s. If r = 1/3 when s = 36, find

a). r when s = 9, b). s when r = 1/4.

8). If y is inversely proportional to the square of x and x = 5 when y = 2,find y when x = 10.

9). y varies inversely as the square root of x and y = 5 when x = 9.Calculate the value of y when x = 25.

10). y α 1/√x and y = 2 when x = 25.Calculate the value of x when y = 4 and the value of y when x = 16.

11). If y varies inversely as the square of x and y =18 when x = 2.Find the value of y when x = 3.

12). h varies inversely as the square of r, and h =18 when r = 4.Calculate the value of h when r = 12 and the value of r when h = 8.

13). y and x are connected by the inverse square law ie y α 1/x2 and y = 3 when x = 1.Find the value of y when x = 9 and the value of x when y = 27.

Sketch Graphs. Direct Proportion The Square Law Inverse Proportion The Inverse Square Law

y = kx y = kx2 y = k y = k x x2

yy y

x

y

x x x


Direct and Inverse Proportion.

1). P is directly proportional to Q and P = 6 when Q = 20. Finda). P when Q is 70,b). Q when P is 12.

2). y varies as the square of x and y = 75 when x = 5. Finda). y when x = 1/

2,

b). x when y = 1/3.

3). y is inversely proportional to the square root of x and y = 2 when x = 4. Finda). y when x = 9,b). x when y = 1.

4). Here are 3 graphs that show a relationship between variable "x" and "y".Which of these correspond to which graph ?

a). y and x are in direct proportion.b). y and x have inverse proportion.c). y varies as the square of x.

5). Here are 5 situations where a variable "P" increases with variable "Q". For each one statewhether P and Q are in direct proportion.a). P is the weight of a baboon in pounds, Q is the weight the baboon in Kilograms.b). P is the temperature in Celsius, Q is the temperature in Fahrenheit.c). In the triangle the base is 10 cm. P is the height

of the triangle, Q is the angle marked.d). P is the age in years of man, Q is the height of the man.e). P is the circumference of a circle, Q is the radius of the circle.

6). Jane invests €500 in an account that repays at a flat rate of 4% Simple Interest per annum.a). Describe the type of proportion between the variables T and I, where T is the time in

years and I is the total interest earned.b). Write the equation that connects T and I.

7). A rectangle has area = 450 cm2. L is the length and W the width.a). Describe the type of proportion between L and W.b). Write the actual equation that connects L and W.

8). Den has a counter attached to his bicycle wheel that records the number of revolutions thewheel has made during a journey. n is the number of revolutions and t the total distance.a). What kind of variation is there between n and t ?b). If 12 revolutions of the wheel gives a distance of 34 metres, how far has he cycled if

the total revolutions count is 1350 ?c). What is the actual equation that connects n and t ?d). What is the radius of the bicycle wheel correct to the nearest cm ?

9). The extension, x cm, of an elastic string varies directly with the force, F newtons.If a force of 4 newtons gives an extension of 6 cm, finda). F when x is 48 cm,b). x when F is 14 newtons.

y

x

y

x

y

x

A. B. C.

PQ


10). Michelle draws up a table for polygons. The values shown are n, the number of sides and A the size of the exterior angle.

n 3 4 5A 120 90 72

a). What kind of proportion is there between n and A ?b). What is the actual equation that connects them ?

11). a). G and V are 2 variables in direct proportion.When V = 4, G = 50. Calculate the value ofi). G when V = 20, ii). V when G = 131 1/

4.

b). H and N are 2 variables in inverse proportion.When N = 25, H = 24. Calculate the value ofi). H when N = 5000, ii). N when H = 12.5.

12). The diameter of a bubble is inversely proportional to the external pressure. A bubble hasa diameter of 5 cm when the external pressure is 1000 millibars.a). What is the external pressure when a bubble has a diameter of 4 cm ?b). The external pressure is 2500 millibars. What is the diameter of the bubble ?

13). The following table shows some values of variables x and y, which are linked by theequation

y = 3xn. x 0 1 3 y 0 3 243

Find a). n b). y when x = 5, c). Express x in terms of y.

14). The extension, e, produced in a stretched spring varies directly as the tension, T, in thespring. A tension of 8 units produces an extension of 2.5 cm.a). What will be the extension produced by a tension of 20 units ?b). The spring is extended 13.6 cm, what was the tension applied to the spring ?

15). If y varies directly as the cube of x and y = 24 when x = 2, finda). the value of y when x = 1/

2,

b). the value of x when y = 1029.

16). The electrical resistance (R1 ohms) of a cylindrical piece of wire 1 unit long is inversely

proportional to the square of the radius (x mm). When x = 2 mm, R1 = 2 ohms, find R

1 if

a). x = 1 mm, b). 2.5 mm, c). 8mm, d). 12 mm.

17). The intensity of illumination, I, of a bulb varies inversely as the square of the distance, d. If theintensity of illumination is six units at a distance of 4 m, what is the intensity at a distance of 2m ?

18). The frequency, f kiloHertz (kHz), of the signal transmitted by a radio station is inverselyproportional to its wavelength, w metres. BBC Radio 5 can be heard on wavelength 433mwith frequency 693 kHz.a). Find and equation linking f and w.b). BBC Radio Wales is on frequency 882 kHz. What wavelength is this ?c). Virgin is on frequency 1215 kHz. What wavelength is this ?d). World Service is broadcast on frequency 463 m. What frequency corresponds to this ?e). Talksport is broadcast on frequency 285 m. What frequency corresponds to this ?


Indices (Powers).

Negative Powers

A. Copy and complete these patterns.Write down all the patterns you can see.

1). 7 3 = 7 x 7 x 7 = 343 2). 3 3 = = 277 2 = 7 x 7 = 3 2 = 3 x 3 =7 1 = 7 = 7 3 1 = =7 0 = 1 = 1 3 0 = 1 = 17 -1 = 1 = 3 -1 = 1 =

7 1 3 1

7 -2 = 1 = 3 -2 = =7 2

B. Evaluate the following. Leave in fraction notation where appropriate.

1). 3 -1 2). 2 -3 3). 3 -2 4). 6 -2 5). 4 -3

6). 7 -2 7). 5 -3 8). 6 -1 9). 8 -2 10). 10 -3

11). 2 -4 12). 4 -2 13). 9 -2 14). 10 -5 15). 11 -1

16). 1 -2 17). 1 -3 18). 1 -2 19). 3 -2 20). 5 -2

3 2 5 2 321). 3 -4 22). 4 -3 23). 4 -2 24). 2 -2 =≠25). 9 -3

2 5 3 7 726). 1 -4 27). 2 -3 28). 3 -3 29). 3 -4 30). 2 -5

4 5 4 5 3

Powers representing Roots.

We know that √ 2 x √ 2 = 2 1. In terms of powers that would be 2 x x 2 x = 2 1.From the addition rule of indices the only value x can be is 1/

2. So √ 2 = 2 1/2 .

Show 3√ 4 can be written as 4 1/3 .

The power 1/2 means √ , the power 1/

3 means

3√ etc.

Evaluate the following. Use the calculator key x1/y . This is marked x√ on some calculators.

1). 64 1/3 2). 25 1/2 3). 9 1/2 4). 36 1/2 5). 125 1/3

6). 49 1/2 7). 8 1/3 8). 4 1/2 9). 27 1/3 10). 81 1/2

11). 144 1/2 12). 2744 1/3 13). 1296 1/214). 343 1/3 15). 205379 1/3

16). 576 1/2 17). 729 1/3 18). 512 1/3 19). 16 1/4 20). 625 1/4

21). 1 1/2 22). 81 1/4 23). 16 1/2 24). 216 1/3 25). 830.584 1/3

26). 0.25 1/2 27). 0.027 1/3 28). 0.64 1/2 29). 2.197 1/3 30). 42.25 1/2

Negative and Harder Fractional Powers.

A). Evaluate the following. Leave in fraction notation where appropriate.

1). 9 -1/2 2). 64 -1/2 3). 64 -1/3 4). 125 -1/3 5). 81 -1/2

6). 144 -1/2 7). 2744 -1/3 8). 343 -1/3 9). 169 -1/2 10). 16 -1/4

11). 49 -1/2 12). 512 -1/3 13). 3375 -1/3 14). 676 -1/2 15). 24 -1

( )

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )


16). 4 -1/2 17). 27 -1/3 18). 343 -1/3 19). 144 -1/2 20). 1296 -1/2

9 125 125 81 421). 25 -1/2 22). 343 -1/3 23). 1331 -1/3 24). 81 -1/4 25). 225 -1/2

16 8 512 16 16926). 144 -1/2 27). 125 -1/3 28). 121 -1/2 29). 343 -1/3 30). 205379 -1/3

4 64 49 216 2744

B). The power 3/2 can be split in to two parts. 1/

2 x 3/

1 .

This means we would square root the number first then cube the resulting number.

Note. When deciding which part to work out first, always try to make the number smaller ratherthan larger. Smaller numbers are easier to deal with!

E.g. Evaluate 36 3/2 = 6 3 = 216

Evaluate the following. Leave in fraction notation where appropriate.

1). 4 3/2 2). 27 2/3 3). 9 3/2 4). 16 3/2 5). 8 2/3

6). 27 4/3 7). 81 3/2 8). 125 2/3 9). 16 5/4 10). 64 4/3

11). 343 2/3 12). 216 5/3 13). 81 3/4 14). 256 3/4 15). 169 3/2

16). 4 -3/2 17). 81 -3/2 18). 27 -2/3 19). 8 -2/3 20). 27 -4/3

21). 343 -2/322). 121 -3/223). 9 -5/2 24). 81 -3/4 25). 16 -5/4

C). Evaluate the following. Leave in fraction notation where appropriate.

1). 9 3/2 2). 81 3/2 3). 9 3/2 4). 9 3/2 5). 144 3/2

16 49 4 81 1006). 27 2/3 7). 343 2/3 8). 216 2/3 9). 81 3/2 10). 64 3/2

8 64 8 9 4911). 121 -3/2 12). 64 -2/3 13). 27 -2/3 14). 64 -2/3 15). 512 -4/3

169 27 125 216 2716). 125 -5/3 17). 64 -5/3 18). 1296 -1/419). 2401 -3/4 20). 81 -3/4

64 729 256 6561 62521). 216 -2/3 22). 625 -3/4 23). 243 -1/5 24). 729 -5/3 25). 81 -5/4

64 16 3125 1728 1296

Missing Values.Find the value of x.

1). 7 x 7 x 7 = 7 x 2). 5 x x 5 4 = 5 6 3). 3 x = 9 4). 4 2 x 4 x = 4 5

5). x 2 = 36 6). 4 3 = x 7). 2 x = 16 8). 4 x = 649). x 3 = 27 10). 9 4 ÷ 9 x = 9 3 11). 7 x ÷ 7 4 = 7 5 12). (4 x ) 2 = 4 4

13). (5 2 ) 3 = 5 x 14). 3x = 81 15). (2 4 ) x = 2 12 16). 4x = 25617). 57 = x 18). x2 = 121 19). 2x = 128 20). 3x = 24321). x 3 = 1000 22). 7 x = 2401 23). 6 x = 216 24). 13 x = 16925). 25x = 5 26). 64 x = 4 27). 121 1/2 = x 28). 625x = 529). 81x = 9 30). 32x = 2 31). 641/2 = x 32). 64x = 233). 125x = 1 34). 64x = 1 35). 343x = 1 36). 256x = 1

5 8 7 437). 7 1/2 x 7 1/2 = 7 x 38). 6 x x 6 x x 6 x = 6

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( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( )( )

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Indices 1.

The table shows powers of 2 to a value of 16 384.Some have been filled in to show how these numbers can

be expressed as powers of 4, 8, 16 etc.Do not cancel down fractions.

1). Copy and complete the table.

Powers of No 2 4 8 16 32 64 128 256

2 21 81/3 641/6 2561/8

4 22 41

8 23

16 42

32

64 82 641

128 1281

256 162

512

1024 322

2048 211

4096

8192

16384 47

2). Write out any number patterns you notice.


Indices 2.

The table shows powers of 3 to a value of 4 782 969.Some have been filled in to show how these numbers can

be expressed as powers of 3, 9, 27 etc.Do not cancel down fractions.

1). Copy and complete the table.

Powers of No 3 9 27 81 243 729 2187 6561

3 2431/5 65611/8

9 812/4

27

81 34

243

729

2187 21871

6561

19683 39 273

59049

177147

531441 96 7292

1594323

4782969 314

2). Look at the answers in the last table (Indices 1), what do you notice ?

3). Draw another table similar to the one above based ona different power of your choice.


Compound Growth and Decay 1.(Exponential Functions).

1). Some organisms reproduce by splitting in two.The diagram shows how one organism reproduces.At the end of every hour every existing organism splits in two.a). Copy the diagram and draw the next 2 hours.b). How many organisms will there be after

i). 6 hours, ii). 9 hours, iii). 14 hours ?c). Plot a graph of the number of organisms for the first

10 hours.d). How many organisms will there be after n hours ?e). After how many hours will the population be more than

i). 10 000, ii). 500 000, iii). 1 000 000 ?

2). Another organism reproduces by splitting in three every hour.i.e. at the end of every hour every existing organism splits in three.a). Draw a diagram showing the first 5 hours.b). How many organisms will there be after

i). 3 hours, ii). 6 hours, iii). 10 hours ?c). Plot a graph of the number of organisms for the first 8 hours.d). How many organisms will there be after n hours ?e). After how many hours will the population be more than

i). 10 000, ii). 500 000, iii). 1 500 000 ?

3). Another organism reproduces by splitting in five every hour.a). Draw a diagram showing the first 5 hours.b). How many organisms will there be after

i). 4 hours, ii). 7 hours, iii). 10 hours ?c). Plot a graph of the number of organisms for the first 8 hours.d). How many organisms will there be after n hours ?e). After how many hours will the population be more than

i). 10 000, ii). 500 000, iii). 2 000 000 ?

4). A very generous Building Society offers an interest rate of 30% p.a. paid at the end ofeach year. Benny puts €1000 into the Building Society and never takes any money out.a). Copy and complete the table below.

Time 0 1 2 3 4 5 6 7 (Completed Years)

Amount in Building Society 1000 1300 (€’s)

b). Plot a graph showing the amount in the Building Society for the first 8 years.c). After how many years would he have more than

i). €8000, ii). €10 000, iii). €20 000 ?d). By what factor would you multiply the amount of money in the Building Society

to calculate the next years amount ? (Indicated by arrows on the table).e). How much money would be in the building Society after n years ?

Number of hours0 1 2 3


A quantity that grows by being multiplied by the same amount in equal intervals of time is calledexponential growth. A quantity that grows when the same amount is added in equal intervals of timeis called uniform growth. Try this investigation to understand this more.

Annual Bonus Schemes.Which of the following Annual Bonus Schemes would you prefer ?A. €1000 now, €900 the year after, €800 the year after that and so on.B. €100 now, €200 the year after, €300 the year after that and so on.C. €100 now, 1.5 times as much next year, then 1.5 times the previous years total, and so on.D. €10 now, €20 next year, €40 the year after, €80 the year after that and so on.

Exponential decay is when the multiplier used is less than 1.

5). The value of a certain make of motor bike depreciates by 18% every year after it has been bought. It initially cost €10 000. a). Copy and complete the table below.

Time 0 1 2 3 4 5 (Completed Years)

Value of Motor bike 10000 8200 (€’s)

b). Plot a graph showing the value of the motor bike over the first 8 years.c). After how many years would the value of the motor bike fall below

i). €8000, ii). €6000, iii). €3000 ?d). By what factor would you multiply the value of the motor bike

to calculate the next years value ? (Indicated by arrows on the table).e). How much money would the motor bike be worth after n years ?

6). The population of a small town falls by 7% every year. Initially the population was 28 000. a). Copy and complete the table below.

Year 0 1 2 3 4 5 Population 28 000

b). Plot a graph showing the population over the first 10 years.c). After how many years will the population fall below

i). 22 000, ii). 18 000, iii). 12 000 ?d). What will the population be after n years ?

7). The activity of a radioisotope falls by 32% every hour.The initial activity is 3500 counts per minute (cpm).a). What will the cpm be after

i). 4 hours, ii). 9 hours, iii). 15 hours ?b). Plot a graph showing the cpm of the radioisotope over the first 10 hours.c). After how many hours will the cpm fall below

i). 2 000, ii). 500, iii). 100 ?

8). Write a formula that will help calculate percentage compound growth and decay.


Compound Growth and Decay 2.(Exponential Functions).

N = No ( 1 ± r ) n

100N = Existing amount at this time, N

o = Initial amount,

r = Percentage change per hour/day/year, n = Number of hours/days/years.

1). A woman invests €4 000 in a building society account which pays 6% per annum.How much will there be after 8 years ?

2). The activity of a radioisotope falls by 22% every hour. If the initial activity is 1800 countsper minute, what will it be after 5 hours ?

3). In a sample of bacteria, there are initially 2500 cells and they increase in number by 12%per day. Find the number of cells after 8 days.

4). A car depreciates in value by 19% each year. If Ronald bought a car 3 years ago for €10500,what is the value of the car now ?

5). A machine was originally bought for €6000. It depreciates by 10% each year.What will it be worth at the end of 4 years ?

6). During the first few weeks of its life, an octopus increases its body weight by 6% each day.An octopus was born with a body weight of 180 grams. How much will it weigh aftera). 1 day, b). 3 days, c). 1 week ?

7). In a sample of bacteria, there are initially 3400 cells and they decrease in number by 32%per day. Find the number of cells aftera). 1 day, b). 3 days, c). 5 days, d). 12 hours ?

8). A man invests €500 in a saving account at an annual rate of interest of 3%. He makes nofurther deposits or withdrawals. Compound interest is added each year.a). How much will he have in his account at the end of the second year ?b). After how many complete years will he first have more than €600 in his account ?

9). A flower is placed in a vase. During the course of each day it loses 6% of its water content. It will begin to droop after losing 28% of its original water content.a). What percentage of its original water content will it lose after 4 days ?b). If the flower had drooped after y days, what is the minimum possible value of y?

10). Find the compound interest on €3000 at 6% for 4 years, and the total amount of moneyat the end of that time.

11). An infectious disease is being successfully reduced using antibiotics. In 1996, 5600 peoplesuffered from the disease. It appears that the number of people catching the disease isfalling by 14% each year.a). Find the number of people catching the disease in

i). 1997 ii). 1998 iii). 1999 iv). 2000.b). Find the year in which the number of people catching the disease falls below 2000.


12). Every day a biologist measures the area of an organism growing in her laboratory.Her results are shown in this table.

Time (days) 0 1 2 3 Area (cm2) 3.00 3.54 4.18 4.93

a). Show that the day-to-day percentage increase in area is the same throughout thegrowing period.

b). State the daily percentage increase in area.c). Calculate the area at the end of Day 6.d). Estimate the number of days it takes for the organism to quadruple in area.

13). A company estimates that a car depreciates in value by 22% each year. How much does itestimate that a car it buys for €18400 is worth at the end of 4 years ?

14). Since 1998, the population of the town of Littleton has been falling at a constant rate.The table gives the population statistics (to 3 s.f.) for the years 1998-2000.

Year 1998 1999 2000 Population (1000's) 15.0 13.8 12.7

a). By what percentage is the population of Littleton falling each year ?b). If the trend continues predict the population for 2004 ?c). When is it expected that the population will fall to 50% of its 1998 figure if

this trend continues at this rate ?

15). A biologist measures the number of cells in a culture at the end of each day. Her results,expressed to the nearest thousand, are as follows.

Day 1 2 3 4 5 6 7Number of cells 20000 28000 39000 55000 77000 108000 151000

a). Estimate the number of cells there will be at the end of the next day.b). On what day will the number of cells exceed 400000 ?

16). The activity of a radioisotope falls by 16% every hour.The initial activity is 3500 counts per minute.a). What will it be after

i). 2 hours, ii). 4 hours, iii). 8 hours ?b). After how many hours will the activity drop to below 500 counts per minute ?

17). A colony of ants increase by 12% per week. Initially there are 50 ants.a). How many will there be after

i). 4 weeks, ii). 8 weeks, iii). 13 weeks ?b). After how many weeks will the population of the colony exceed 250?

18). The speed of a ball rolled along short grass falls by 15% every second.If the initial speed was 14 m/s, find the speed after

a). 2 seconds, b). 5 seconds, c). 10 seconds.


Exponential Functions.

Carbon Dating.

Carbon dating is a technique for discovering the age of ancient organic objects. All bodies havethe same proportion of Carbon 14 as occurs in the atmosphere at the time. After death the uptakeof Carbon into the body ceases and hence the amount of Carbon 14 in the body begins to fall bynatural radioactive decay. By finding by how much the level of Carbon 14 has fallen in a samplewe can tell the age of the sample. Carbon 14 is measured in “counts per minute per gram ofcarbon” or cpm. Living bodies have a count of 15.3 cpm.

Hence the amount of Carbon 14 in a sample is given by

a = 15.3 x 0.866t where t is thousands of years since the body died.

1). Find the amount of Carbon 14 in a bodya). 3000 years after it dies, b). 5000 years after it dies,c). 5600 years after it dies, d). 8000 years after it dies,e). 15000 years after it dies, f). 40000 years after it dies ?

2). The half-life is the time it takes for the Carbon 14 to reduce itsquantity by half. What is the half-life of Carbon 14 ?

3). Plot a graph showing how the amount of Carbon 14 in a sample varies over 20 000 years.

4). Use your graph, or otherwise, to find how old samples are that have a Carbon 14 counta). 8.61 cpm, b). 6.18 cpm,c). 5.20 cpm, d). 2.57 cpm.

5). The Dead Sea Scrolls were dated at around 200 BC. How much Carbon 14 would youexpect in a sample of the scrolls ?

6). The paintings in the Lascaux Cave in France were drawn approximately 15 500 yearsago. How much Carbon 14 would you expect in a sample from the paintings ?

7). In the Chauvet Caves in France the animal cave paintings were found to contain0.15 cpm of Carbon 14. Approximately how long ago were these paintings made ?

8). Papyrus sheets were found to contain 10.82 cpm of Carbon 14.Approximately how old were the papyrus sheets ?

One of the most famous incidents of Carbon dating was that of the Turin Shroud in 1988.This was the shroud that supposedly covered Christ’s body in hours and days after thecrucifixion. It was found to be a medieval hoax. The Shroud was dated at AD 1320 ± 65years by three independent laboratories in Arizona, Oxford and Zurich. This in fact ties inwith the historical evidence as the Shroud only came to public notice around 1345

Carbon 14 produces a naturally occurring form of radiation.


Radioactivity.

Radioactivity is measured in pico curries. Radioactivity decays exponentially.The amount of radioactivity at any given time for Strontium 90 is

y = A x 0.976t where A is the initial radioactivityand t is the number of years.

1). If the initial radioactivity of a trace of Strontium 90 is 134 pc, what is the radioactivitylevel aftera). 4 years, b). 8 years, c). 10 years, d). 15 years ?

2). Use trial and improvement, or otherwise, to find the half life for Strontium 90.

For one half-life to occur y and A must be in the ratio 0.5 : 1,hence a general formula must be

0.5 = 1 x numbert

where t is time of the half-life.

3). The half-life of Iodine is 8 days. Find a formula, in the form y = A x numbert , to showthe amount of radioactivity at any given time for Iodine.

4). The half-life of Radium is 1620 Years. Find a formula, such as the one above, to showthe amount of radioactivity at any given time for Radium.

5). Here are the half-lives of 4 more elements.

Element Half lifeUranium 4.5 x 109 yearsStrontium 28 yearsBismuth 19.7 minutesPolonium 0.15 milliseconds

Find a formula in the form y = A x numbert fora). Bismuth, b). Polonium.

Radiometric Dating.

Carbon 14 dating allows relatively young sediment and organic remains to be dated. There areother decay series that can be useful for dating older rocks such as the rubidiun-strontium series.The crucial feature in all these dating procedures is the fact that the decay is exponential, hencecreating half-lives. The half-life of Carbon 14 is 5700 years as we previously calculated. Thehalf-life of the rubidiun-strontium series is an amazing 48 000 million years, this can date theoldest rocks in the world and indeed the solar system. The rubidiun-strontium series has beenused to date an ancient slab of the sea floor, Dongwanzi ophiolite, to be 2500 million years old.The moon was found to be 3700 million (± 70 million) years old and the earth to be 4600 millionyears old.


Surds.Multiplying Surds.A. Express these surds in the form a√b.

1). √8 2). √27 3). √20 4). √32 5). √806). √44 7). √75 8). √72 9). √45 10). √10811). √28 12). √125 13). √245 14). √192 15). √40516). √112 17). √63 18). √180 19). √99 20). √48

B. Express each of the following as the square root of a single number.

1). 3√2 2). 2√5 3). 6√2 4). 4√5 5). 3√36). 2√3 7). 5√5 8). 7√2 9). 6√3 10). 5√1111). 9√3 12). 10√7 13). 8√5 14). 11√3 15). 15√6

C. Work out the following. Leave the answer in surd form where appropriate.

1). √3 x √6 2). √6 x √2 3). √10 x √5 4). √8 x √5 5). √10 x √26). √3 x √3 7). √2 x √8 8). √14 x √2 9). √2 x √9 10). √5 x √1511). √3 x √8 12). √5 x √5 13). √2 x √18 14). √6 x √6 15). √5 x √30

D. Surds in the form a√b can be multiplied.E.g. 3 √3 x 4 √2 = 12 √6

Work out the following. Simplify where possible.

1). 2√7 x 4√3 2). 2√5 x 3√2 3). 2√3 x 3√3 4). 5√3 x 7√25). 4√2 x 5√2 6). 2√3 x 6√5 7). 2√8 x 5√5 8). 2√3 x 3√89). 2√2 x 3√8 10). 2√32 x 3√2 11). 5√6 x 2√3 12). 2√7 x 3√313). 5√5 x 4√4 14). 4√24 x 2√3 15). 5√3 x 4√21 16). 3√5 x 4√2417). 4√8 x 3√5 18). 3√2 x 8√40 19). 4√18 x 2√5 20). 2√6 x 5√42

Dividing Surds.A. Work out the following. Simplify where possible.

1). √10 ÷ √2 2). √20 ÷ √5 3). √150 ÷ √3 4). √90 ÷ √2 5). √72 ÷ √66). √42 ÷ √7 7). √55 ÷ √5 8). √144 ÷ √8 9). √48 ÷ √6 10). √126 ÷ √711). √80 ÷ √5 12). √588 ÷ √3 13). √320 ÷√10 14). √1050 ÷√3 15). √320 ÷ √5

B. Surds in the form a√b can be divided.E.g. 8√6 ÷ 2√3 = 4√2

Work out the following. Simplify where possible.

1). 6√15 ÷ 2√3 2). 14√3 ÷ 7√3 3). 8√6 ÷ 2√3 4). 20√15 ÷ 4√55). 10√2 ÷ 2√2 6). 15√7 ÷ 3√7 7). 10√30 ÷ 5√5 8). 18√32 ÷ 3√89). 27√24 ÷ 3√8 10). 24√28 ÷ 3√2 11). 2√27 ÷ 2√3 12). 21√3 ÷ 3√313). 32√35 ÷ 4√5 14). 10√24 ÷ 2√3 15). 8√32 ÷ 4√2 16). 4√48 ÷ 4√317). 2√10 ÷ 2√2 18). 12√28 ÷ 3√7 19). 15√30 ÷ 5√5 20). 30√150 ÷ 5√6

Answers


Mixed Questions.A. Work out the following. Simplify where possible.

1). (√2)3 2). (√3)3 3). (√2)5 4). (√3)4 5). (√5)5

6). (3√2)2 7). (2√7)2 8). (2√3)3 9). (2√2)3 10). (2√3)2

11). (2√5)2 12). (5√3)2 13). (2√5)3 14). (3√6)2 15). (3√5)3

B. Given that √2 = 1.41, √3 = 1.73 and √5 = 2.24 find the values of each of the following:

1). √18 2). √8 3). √48 4). √12 5). √756). √20 7). √32 8). √27 9). √50 10). √4511). √72 12). √98 13). √108 14). √80 15). √12516). True or false ? To calculate the square roots of all the whole numbers from 1 to 100

you only need the square roots of all the prime numbers between 1 and 100.

C. Simplify

1). √8 + √2 2). √20 - √5 3). √3 + √ 12 4). √8 - √ 25). √27 + √12 6). √125 - √20 7). √48 + √75 8). √18 + √729). √75 - √27 10). √80 - √20 11). √108 - √27 12). √27 - √1213). √147 - √108 14). √48 - √27 15). √98 + √8 + √2 16). √99 - √44 - √1117). 3√2 - √18 18). √175 - 4√7 19). 3√8 + √50 20). 5√5 + √2021). 2√45 + 3√20 22). 3√32 - 2√18

D. Eg. Rationalise 4 = 4 x √2 = 4√2 = 2√2 √2 √2 x √2 2

Rationalise means make the denominator a rational number.Rationalise

1). 3 2). 10 3). 21 4). 8 5). 24√3 √5 √7 √2 √6

6). 1 7). 1 8). 1 9). 2 10). 9√3 √2 √5 √3 √15

11). 21 12). 8 13). 2 14). 9 15). 30 √6 √18 √5 √6 √75

16). √12 17). √12 18). 3√2 19). 3√7 20). 4√5√50 √ 3 √10 √21 √20

Eg. Rationalise 1_ = 1 x (√7+√5) = √7+ √5= √7 + √5 √7 - √5 (√7-√5)(√7+√5) 7 - 5 2

Eg. Rationalise 1_ = 1 x (√2-1) = √2 - 1 = √2 - 1 √2 + 1 (√2+1)(√2-1) 2 - 1

21). 1__ 22). 1_ 23). 4_ 24). 6_ 25). 4_ √5 + √2 √3 - √2 √7 + √5 √13 - √7 √5 + √326). 7__ 27). 4_ 28). 12_ 29). 6_ 30). 6_ √3 + 2 √11 - 3 √7 + 3 √13 - 2 √24 - √6


Rational and Irrational Numbers 1.

Rational Numbers.

Integers mean whole numbers.The Natural Numbers are 1, 2, 3, 4... or the positive integers.

A rational number is any number that can be written as a where a and b are integers. b

Rational numbers can be located exactly on the number line.These numbers are rational:

-3 -11/2

0 3/4

3 51/2

6.4 as they can be written in the form a/

b

i.e. -3 -3 0 3 3 11 64 1 2 1 4 1 2 10

A. All the following numbers are rational. Write them in the form a/b.

1). 7 2). 4 3). -3 4). -8 5). 146). 12/

37). 24/

58). -31/

99). -11/

710). 82/

3

11). 34/5

12). -16/13

13). 45/7

14). 83/8

15). 57/15

16). 0.4 17). 2.6 18). -3.5 19). 1.7 20). 2.121). 4.13 22). 0.04 23). 6.73 24). -1.462 25). 3.743526). -4 27). -1.495 28). 72/

529). 6.014 30). -37/

8

0.3 means 0.33333.... 0.14 means 0.14141414.. 0.216 means 0.216216216.....We have met notation for recurring decimals before.

All numbers that end in an exact decimal or are a recurring decimal are rational numbers.

B. Change each of the following fractions to a decimal.Give the answer as a terminating or recurring decimal as appropriate.

1). 1/8

2). 1/3

3). 1/12

4). -5/8

5). 31/7

6). 22/3

7). 44/5

8). -31/9

9). -11/13

10). 84/11

11). 7/13

12). 21/15

13). -7/20

14). -9/4

15). 45/18

16). -14/21

17). 123/30

18). 41/12

19). 25/18

20). -14/8

C. Change these fractions to decimals and then put them in ascending order.

1). -11/4

4/9

27/8

-5/12

2). 32/9

-12/9

-24/5

25/6

3). 11/7

11/9

11/10

12/21

4). 28/13

31/14

23/7

20/9

5). -14/9

-11/2

-11/3

-15/8

6). 16/5

25/8

19/6

26/7

7). -15/6

15/9

-18/11

11/9

8). -15/12

-17/9

18/15

-11/9

9). -3/5

4/7

-1/6

9/16

10). 4/9

-5/11

-6/13

1/2

..... ..... ..... ..... .....


E.g.1 Convert 0.5 to a fraction.

Let x = 0.555555....Therefore 10x = 5.555555.... (Why have we multiplied by 10 ?)

Now subtract the two lines 10x = 5.555555....x = 0.555555.... -9x = 5

Therefore x = 59

E.g.2 Convert 0.123 to a fraction.

Let x = 0.123 123 123....Therefore 1000x = 123.123.123 123.... (Why have we multiplied by 1000 ?)

Now subtract the two lines 1000x = 123.123.123 123.... x = 0.123 123 123.... - 999x = 123

Therefore x = 123 = 41999 333

D. Show these recurring decimals are rational numbers.

1). 0.4 2). 0.6 3). 0.8 4). 0.45 5). 0.616). 0.8 7). 0.416 8). 0.26 9). 0.09 10). 0.111). 0.636363... 12). 0.215215... 13). 0.155555.. 14). 1.33333...15). 4.3272727... 16). 0.191919... 17). 0.545454... 18). 0.818181...

19). 0.31285285285... 20). 0.142857142857142857...

E. Show the following are rational numbers by writing them in the form a/b.

1). 0.6 2). 0.2 3). 0.125 4). 0.46 5). 0.76). 0.3 7). 0.875 8). 0.9 9). 0.96 10). 0.3411). 0.21 12). 0.75 13). 0.48 14). 0.04 15). 0.65516). 0.4375 17). 0.14 18). 0.85 19). 2.561 20). 0.784

F. Investigate what happens when youmultiply recurring and terminating decimals.Find fractions that are terminating decimals.Find fractions that are recurring decimals.Multiply the fractions together.See if the result is a terminating orrecurring decimal.Repeat this for lots of different fractions.Fill in the results table.

Recurring Terminating decimal decimal

Rec

urri

ng

dec

imal

Term

inat

ing

de

cim

al

.....

..... .....

..... .....

..... ..... ..... .....

.......... .......... .....

..... ..........

..... ..... ..... ..... ..... .....

..... ..... ..... .......... .....


Rational and Irrational Numbers 2.

Irrational Numbers.

Irrational Numbers are numbers that are not rational!!

These numbers exist on the number line but can not be precisely located. These numbers arenon-recurring decimals with an unending string of digits i.e. the decimal part of it goes on and onwithout ever showing any repeating pattern.

Here are some numbers that are irrational:

1). Square roots of numbers that are not perfect squares.E.g. √2 √3 √5 √6

2). Cube roots of numbers that are not perfect cubes.E.g.

3√2

3√3

3√4

3√5

3). π and expressions involving π.E.g. π π2 π + 3

4). Most of the trig. functions where the angle is a rational number of degrees.E.g. cos 17˚ tan 32.5˚ sin 38˚

(Beware the ones such as tan 45˚, sin 30˚, cos 60˚ etc.)

A. Say whether these are rational or irrational numbers.

1). 2.777... 2). √3 3). 3π 4). 0.45732 5). √176). √16 7). 4√5 8). 1 + √7 9). π + 4 10). tan 47˚11). 0.666... 12).

3√7 13). √11 - 2 14).

3√64 15). 1

16). sin 23˚ 17). π2 18). cos 90˚ √9

B. Write an irrational number that lies between

1). 2 and 3 2). 3 and 4 3). 4 and 5 4). 5 and 6 5). 6 and 76). 7 and 8 7). 8 and 9 8). 9 and 10 9). 10 and 11 10). 11 and 12.

C. What can be added to each of the following to make a rational number ?

1). √5 2). 3√2 3). π + 4 4). tan32˚ 5). √3 - √76). 3π 7). 4 + sin16˚ 8). 5√2 9). 7 - π 10). √2 - 1.4

D. What can the following be multiplied by to make a rational number ?

1). 1 2). √3 3). 4π 4). 2__ 5). √5π cos72˚ 6

6). π2 7). tan12˚ 8). √2 9). 7 10). 3√36 5 √3 √8 π

Easy !!


Mixed Questions.

1). Which of A, B, and C are rational. (Give a reason for your answer).A = 1.4534534.... B = 1.4537623.... C = 1.45376237

2). a). What could be added to 5 + √2 to give an answer which is rational ?b). State whether the following are rational or irrational.

i). √5 ii). 3π iii). 22/7

iv). 1.66666.....

3). Q is an irrational number.a). Is 2Q rational or irrational ? Give a reason for your answer.b). The lengths of the sides of a right angled triangle DEF are √3 cm, √5 cm and √7

cm. The area of this triangle is A cm2. Is the value of A rational or irrational ?Show your working.

4). a). Write down an irrational number that lies between 7 and 8.b). Q is a rational number which is not equal to zero. Show clearly why 1

must also be rational. Q

5). a). Write 1.32 as a rational number in the form a , where a and b are whole numbers. b

Given that n = 1.8 (read as 1.888.....) ,b). i). write down the value of 10n,

ii). hence write down the value of 9n,iii). express n as a rational number in the form a , where a and b are whole

numbers. bc). Write down an irrational number which has a value between √11 and √12.

6). a). What is a rational number ?b). Show that 0.15555.... is a rational number by writing it in the form a ,

where a and b are whole numbers. b

7). F = 0.16a). Write down the value of

i). 100F,ii). 99F,iii). F ( written as a rational number in the form a where a and b are whole numbers). b

b). Use a similar method to write 0.129 as a rational number in the form a/b.

8). p is rational and q is irrational. State if these are rational or irrationala). p + q, b). pq.

9). Write an example showing how the product of two irrational numbers can be rational.

10). Solve these equations. State whether the solution is rational or irrational.a). 3x + 4 = 14 b). 2x - √5 = √10 c). x2 + 3 = 5d). x2 - 1 = 8 e). √(9x2) = 12 f). √x - 2 = 5

..... .....

..... .....

.....

Level 9/10. Pack 1. Page 33. [email protected]

Using Surds in Trigonometry

ExampleIf sin x = ¾, find tan x and cos x.

This can be done without calculating the angle x. Just remember

sin oppositexhypotenuse

= So, we could draw a triangle like this:

We can now use Pythagoras’ theorem to calculate AB.

AB2 = 42 - 32

= 7AB = √7

So,3tan7

x = and7cos

4x =

ExerciseCopy and complete the table below using a similar method to the one above.Give all answers in surd form.

Sin x Cos x Tan x35

13

52

72

1 5

2

13 5 2

7 3

4 3 9

3 + 1

8 - 5

x 3

4

A B


This is a right-angled, isosceles triangle.Two sides have length 1 unit.

Use Pythagoras’ theorem to find the length of thehypotenuse AB. Leave your answer as a surd.

Using surds where necessary, write down the ratios for sin 45º,cos 45º and tan 45º.

Task 2Here is an equilateral triangle. Each side is 2 units in length.Find the height of the triangle (in surd form) using Pythagoras’theorem and use this to express the sine, cosine and tangent of60° and 30°.

2 2

2

Trigonometric Ratios for 45º, 30º and 60ºin Surd Form

1

1

C B

A

Check your answers using a calculator.

Task 1

Task 3

30º

15º

A

B

C

P

In triangle ABC, AC = BC = 1 unit.

Use your value of tan 30º from task 2 to find thelengths of CP and BP.

Show that AP = 23 units.

Using the Sine rule for triangle ABP, show that

3 1sin15º2 2

−=

Task 4This is a unit octagon. Each side is 1 unit.

Show that ∠BAC = 67½º and ∠ABC = 22½º

Try to find the tangent, sine and cosine of 67½º and22½º in surd form.

B

C A

Use the value of sin 15º to find cosine and tangent of 15º in surd form.


Using Trigonometric ratios for 45º, 30ºand 60º in Surd Form

Find the missing length marked x in each diagram. Express your answers in surd form.

1. 2. 3. 4.

45º

8 cm

x

x

5cm 60º

30º

x

16 cm

45º

x

7 cm

5. 6. 7. 8.

x 3 cm

30º

60º

10m

x

45º

√5 cm

x

x

√7 cm

60º

9. 10. 11.

60º

45º

3 cm

x

x

30º 45º

5 cm

45º

30º

x

13 cm

12. 13.

45º 45º

13 cm x

45º

45º

30º 4 cm

x

15.

14.

60º 30º

9 cm

x

16.

30º

45º

4√3

x

x 30º 45º

45º

11 cm


A

B b

a

A : B = a : b

Show that the length and width of the paper must be in the ratio 1: √2.

Investigations with Irrational and Prime Numbers

A4 paper

A4 paper (foolscap) is designed so that if it is cut in half as shown, the resulting rectangle ismathematically similar, that is, the sides remain in the same ratio.

Paper Folding

1. Take a piece of A4 paper and fold it to make an isosceles triangle like this:

Can you prove that two sides are equal?

Show that the perpendicular height of the isosceles triangle is 7

2.

2. Fold another piece of A4 paper to make a kite as shown.

Use your knowledge of right-angled triangles and similar triangles to show thelengths of the adjacent sides are equal.


Sine Rule Surd Problem

In triangle ABC, 2sin

3A = and

3sin4

B = .

Show that 21 26sin

12C += .

(Hint: drop a perpendicular line from C onto AB.)

A B

C

Paper Folding continued

3. Try this method for folding an equilateral triangle from a sheet of A4 paper.

Can you prove that the triangle is equilateral?

4. This is how you fold a regular pentagon from A4 paper.

Prove that the shape is a regular pentagon.

Recurring or Terminating?15 is a terminating decimal, 0.2.

38 also terminates, 0.375.

16 produces a recurring decimal, 0.1666...

Which numbers produce terminating decimals and which produce recurring ones ?


Proving IrrationalityThe word rational is related to ratio. A rational number can be expressed as a ratio (or fraction)of two integers. We can prove that a number is irrational by assuming first that it is rational andcould be expressed as a fraction, then showing how this would lead to something that cannot betrue. This leads to the conclusion that the number cannot be rational after all, so it is irrational.This technique is usually referred to as proof by contradiction.

Proof that √√√√√2 is irrational.Firstly, assume that √2 is rational and can be expressed as a fraction.

So, 2 ab

= where a, b are integers with no common factors, that is,

the fraction is cancelled to its lowest terms and a, b ≠ 0.

Squaring both sides:2

22 ab

=

which leads to, 2 22a b=

This means that a2 must be a multiple of 2 and hence, a is a multiple of 2. We could say;

a = 2n

By substituting 2n for a, 4n2 = 2b2

which leads to, b2 = 2n2 which means b2, and hence,values of b are multiples of 2.

This is our contradiction. If a and b are both multiples of 2 then they have a commonfactor of 2. This contradicts the assumption that a and b had no common factors.The assumption that √2 is rational is contradicted. Therefore, √2 must be irrational.

Using proof by contradiction, prove that the following numbers are also irrational.

a). √3 b). √5 c). √12 d). √10 e). 3 10

More Recurring or Terminating?

15 is a terminating decimal, 0.2.

215

is also a terminating decimal, 0.04.

13 is a recurring decimal, 0.333...

213

is also a recurring decimal, 0.111...

Is the square of a terminating decimal always a terminating decimal, and is the square of arecurring decimal always a recurring decimal ?


Calculating πππππ

James Gregory (1638 – 1675) wrote about a series for calculating the size of an anglegiven its tangent, t.

3 5 7 91tan ( ) ...

3 5 7 9t t t tt t− = − + − +

The angle is calculated in radians (180º = π radians)

Now we know that tan 45º (or π/4 radians) is 1.So if we substitute t = 1 into Gregory’s series it will return a value for π/4.We can then multiply by 4 to find π.

The series looks surprisingly simple.The more terms you calculate, the more accurate your value of π will be.

1 1 1 1 1tan (1) 1 ...4 3 5 7 9

− π= = − + − +

Try adding the first 5 terms of the series and multiplying by 4. How accurate is your estimate?

Improve this estimate by using a spreadsheet.

The nth term of the series is1( 1)

2 1

n

n

−−−

Notice that 2n-1 is the odd numbers and (-1)n-1 alternates the signsbetween positive and negative.

You can set up your spreadsheet like the one below.Remember * means “multiplied by” and ^ means “to the power of”.

A B C D E F1 n sign Odd number Term Total PI2 1 =(-1)^(A2-1) =2*A2-1 =B2/C2 =D2 =E2*43 =A2+1 =D3+E2

If you Fill Down the formulae in each column you should see increasingly better estimatesfor π in column F. However, you will see this series is a slow method of finding π.

How many terms must you add to find π correct to 2 decimal places ?

45º

1

1


A quicker alternative is by considering 30º or π/6 radians. This leads to this approximation:

2 3 4

1 1 1 12 3 1 ...3 3 5 3 7 3 9 3

π = − + − + − × × × ×

The nth term of the sequence in the brackets is1

1

( 1)3 (2 1)

n

n n

−

−

−− (check why).

Set up a new spreadsheet like the one below.

A B C D E F G1 n sign Power of 3 Odd number Term Total PI2 1 =(-1)^(A2-1) =3^(A2-1) =2*A2-1 =B2/(C2*D2) =E2 =2*SQRT(3)*F23 =A2+1 =E3+F2

See how quickly this series converges.You should be able to get π correct to 15 significant figures in less than 30 lines.

Pierre de Fermat’s Primes 1

One of this French mathematician’s discoveries was that many prime numbers could be expressedas the sum of two square integers.For example,

41 = 42 + 52 and 53 = 22 + 72

How many prime numbers below 100 can you do this for?

Pierre de Fermat’s Primes 2

An odd prime number can be expressed as the difference of two square integers.For example,

11 = 62 - 52 and 17 = 92 - 82

Prove that the two integers must be consecutive and that their sum is equal to the prime number.

Rationalizing

Can you find:

a). Two irrational numbers whose sum is a rational number ?b). A pair of irrational numbers (not the same) with a product that is rational ?c). An irrational number between 3.0 and 3.1 ?d). The number which multiplies (1 + √2) to make a rational number ?


Approximating √√√√√2 by Iteration

First you need to rearrange the equation x = √2 to make an iterative formula that converges.

(square both sides) 2 2x =

(add x to both sides) 2 2x x x+ = +

(factorise left-hand side) ( 1) 2x x x+ = +

(divide both sides by x + 1)2

1xx

x+=+

So your iterative formula is: 12

1n

nn

xxx+

+=+

where xn is your previous solution for x, and xn+1 is your next, improved solution.

Copy the table below and extend it to find √2 correct to 5 decimal places.

xn 2 + xn xn+ 1 xn+ 1 = 2 + xn xn + 1

1 3 2 1.51.5 3.5 2.5 1.41.4

Try to develop an iterative formula for approximating √3.

Each term of this sequence produces increasingly better estimates of √2.

1 3 7 17 41 99, , , , , , ...1 2 5 12 29 70

Each term is calculated using the last term.

If the last term is ab , then the next one is

2a ba b++ .

What value of √2 does the 10th term produce ?

Show why this sequence is actually the same as the iterative formula for √2.


Powers of 10

100.5 = 3.16227766 (9 s. f.)

101.5 = 31.6227766 (9 s. f.)

102.5 = 316.227766 (9 s. f.)

Can you explain why each answer is 10 times bigger than the previous one ?

Investigate other decimal powers of 10 using a spreadsheet.Set up your spreadsheet like the one below. Notice on a spreadsheet ^ means “to the power of”.

A B C1 n Power of 102 0 =10^A23 =A2+0.1

Fill down the formulae in columns A and B. In column A the n numbers increase from zero insteps of 0.1. Column B contains the values of 10n.

Can you solve this equation? 10n = 17Try changing the starting value of n and the step size between each number, n.

LogarithmsThe values you created in the spreadsheet make a very basic table of logarithms.Logarithms (or logs) can be defined like this:

The log of a number (in base 10) is the power to which 10 must be raised to give that number.

If log A = x then 10x = A

For example, log 1000 = 3 because 103 = 1000

The solution to the equation 10n = 17 is, in fact log 17.

Below is a table of all the logarithms of numbers 1 to 50 (3 d.p.)

n log n log n log n log n log1 0.000 11 1.041 21 1.322 31 1.491 41 1.6132 0.301 12 1.079 22 1.342 32 1.505 42 1.6233 0.477 13 1.114 23 1.362 33 1.519 43 1.6334 0.602 14 1.146 24 1.380 34 1.531 44 1.6435 0.699 15 1.176 25 1.398 35 1.544 45 1.6536 0.778 16 1.204 26 1.415 36 1.556 46 1.6637 0.845 17 1.230 27 1.431 37 1.568 47 1.6728 0.903 18 1.255 28 1.447 38 1.580 48 1.6819 0.954 19 1.279 29 1.462 39 1.591 49 1.69010 1.000 20 1.301 30 1.477 40 1.602 50 1.699

Use the table to investigate these statements:a). log AB = log A + log B b). log AB = Blog A

Use the table to find log 60.

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