Level 1 Advanced Mathematics Junior Final Exam 2009 Part …...2009 Level 1 Advanced Math Junior Final Exam (Calculator Section) Page 2 Desmond, King, Nitsch, Normile 1. (5 points)

2009 Level 1 Advanced Math Junior Final Exam (Calculator Section) Page 1 Desmond, King, Nitsch, Normile Version 4 6/15/09

Level 1 Advanced Mathematics Junior Final Exam 2009 Part 2…Calculators are permitted on this section

NAME: _____________________________________________

Teacher: (circle one)

Desmond King Nitsch Normile

Block: A B C D E F G H Instructions Write your name in the space provided at the top of every odd-numbered page.

WRITE ANSWERS IN THE SPACES PROVIDED AND SHOW ALL WORK. Partial credit will not be given if work is not shown. Ask for extra paper if you need it. NOTHING ON THE EXTRA PAPER will be graded unless you explicitly write “see extra paper” on this exam paper AND you clearly indicate the problem number on the extra paper. Make sure your name is on any paper you want graded.

CALCULATORS are permitted. MAKE SURE YOU ARE IN THE CORRECT MODE ON ALL TRIG PROBLEMS. If you use your calculator for something other than numeric calculations, make sure you make clear what your inputs are.

A FORMULA SHEET will be distributed separately. Use the formula sheet for both parts of the exam.

ALL PAPERS: (the exam, extra paper, and the formula sheet) will be collected at the end of the exam.

POINT VALUES: There are 100 points total on this exam, 56 of which are on this calculator section.

Page/max points Score

2. 8

3. 7

4. 13

5. 7

6. 5

7. 10

8. 6

Score Part 2

56

2009 Level 1 Advanced Math Junior Final Exam (Calculator Section) Page 2 Desmond, King, Nitsch, Normile

1. (5 points) City A has a current population of 150,000 and is shrinking at the rate of 3% per year. City B has a current population of 65,000 and is growing at the rate of 5% per year.

a) Write an exponential function

€

A( t) that gives the population of City A as a function of time in years (t = 0 for today).

b) Write an exponential function

€

B( t) that gives the population of City B as a function of time in years (t = 0 for today).

c) What will be the population of City A in 6.5years (nearest integer)?

d) How many years from now will the two cities have the same population (2 decimal places)?

e) What will be the population of each city at that time (nearest integer)?

2. (3 points) A particular house in Lexington had the following value at certain times during the 1990’s.

Year 1990 1993 1995 1998

Value 250,000 364,650 473,850 686,700

a) Use regression to find an exponential function that models the value of the house as a function of time, with t = 0 for 1990. (Round your answer to two decimal places)

Answer: _______________________________________

b) According to your function, what will be the price of a house today, in 2009? Express your answer to the nearest dollar.

Answer: _______________________________________

NAME: _________________________________________


E

F

C

A

Q ( , ) R ( , )

P ( , )O

B ( , )

3. (7 points) Circle O is a unit circle. The m∠AOB = 10˚. The terminal sides of angles ∠AOP ∠AOQ and ∠AOR are located in quadrants II, III, and IV respectively. The reference angle of each is equal to m∠AOB

DEGREE MODE

a) Determine the standard position angle measures of:

∠AOP = _________ ∠AOQ = ________ ∠AOR = ______________

b) Calculate the length of arc ABP. (Express your answer to the nearest 100th.)

ANSWER: ______________________

c) Find the coordinates of points B, P, Q, and R. (Express your answers to the nearest 100th.)

B: _______________________ P: __________________________

Q: _______________________ R: __________________________


4. (4 points) Use algebra to solve the equation for x. Round your answer to the nearest 100th.

€

1+ e2x =12 ANSWER: ________________________________________

5. (4 points) Prove the following identity:

€

sec x − sin x tan x = cos x

6. (5 points) A spacecraft is orbiting around the earth. Its distance above the earth’s surface in kilometers (d) can be approximated by a sinusoidal function of time in minutes (t). At time

€

t = 0 minutes, it is at its highest point,

€

d =1000 km, above the earth’s surface. It is at its lowest point,

€

d =100 km above the surface, 50 minutes later.

RADIAN MODE

a) Using the information given above, find the following:

Amplitude: ____________ Period: ________________

Phase Shift: ____________ Vertical Shift: ____________

b) Write an equation expressing d in terms of t.

ANSWER: ________________________________________________

NAME: _________________________________________


7. (7 points) As the paddlewheel on a steamboat turns, a point on the wheel moves in such a way that its distance in feet, y, from the water's surface is a sinusoidal function of time in seconds given by the equation. RADIAN MODE

€

y = 3+ 7cos π10(x −16)

a) Use an algebraic method to find the first three positive times for which the point is 9 feet from the water's surface. Show all work.

b) Which of the graphs below is the graph of this equation? Circle the letter below the correct graph. THE SCALE ON ALL GRAPHS IS ONE UNIT PER DIVISION

A B

C D

10

8

6

4

2

-2

-15 -10 - 5 5

10

8

6

4

2

-2

- 4

-10 - 5 5 10

10

8

6

4

2

-2

- 4

-15 -10 - 5 5

10

8

6

4

2

-2

- 4

- 5 5 10 15


8. (5 points) A farmer is building a triangular pen for his pigs, as shown in the diagram below. He knows that he wants two sides of the pen to be fencing of lengths 25 ft and 35 ft, and the angle between these sides should be

€

72°.

DEGREE MODE

a) Find total amount of fencing the farmer will need to enclose the pen. Express your answer to the nearest tenth.

ANSWER: ______________________

b) The farmer wants to cover the entire pen with mud. What is the area that will be covered? Express your answer to the nearest tenth.

ANSWER: _____________________________

b

35 ft

25 ft

72°

A

B

C

NAME: _________________________________________


9. (5 points) Two observers, A and B, that are 15 km apart sight an airplane at angles of elevation of °40 and °75 , respectively.

DEGREE MODE

a) How far is the plane from observer A? Express your answer to the nearest tenth.

ANSWER: _____________________

b) Calculate the altitude of the plane. Express your answer to the nearest tenth.

ANSWER: ______________________

10. (5 points) Find each limit using the graph below.

a)

€

limx→3+

f (x) = ______

b)

€

limx→3

f (x) = ______

c)

€

limx→−3

f (x) = ______

d)

€

limx→6+

f (x) = ______

e)

€

limx→6−

f (x) = ______

f)

€

limx→6

f (x) = ______

30 kmA B

C

15 km


10. (6 points) Let f(x), g(x), and h(x) be functions defined by the graph, table, and piecewise formula given below.

Window range [-6, 6] by [-6, 6]

x g(x)

-3 10

-2 7

-1 4

0 1

1 -2

2 -5

3 -8

4 -11

€

h(x) =−x2 if x < 0x2 if x ≥ 0

Find the following values. You do not need to show any work.

a)

€

( f + g)(−2) = __________________

b)

€

(g ⋅ h)(3) = ___________________

c)

€

g( f (5)) = ____________________

d) The graph of k(x) shown below has been transformed from f(x) shown above. Use function notation to write an equation for k(x) in terms of f(x):

Answer: ___________________________

f(x)

NAME: _________________________________________


Level 1 Advanced Math 2009 Final Exam Reference Compound Interest, Exponential Growth

Yearly Compound Interest, Exponential Growth

€

A = P 1+ r( )t

Interest Compounded n times a year

€

A = P 1+rn

nt

Trigonometric Identities

€

tan(x) =sin(x)cos(x)

cot(x) =cos(x)sin(x)

€

csc(x) =1

sin(x)

sec(x) =1

cos(x)

cot(x) =1

tan(x)

1)(cos)(sin 22 =+ xx

)(sec1)(tan 22 xx =+

)(csc)(cot1 22 xx =+

Laws of Sines, Cosines Area of a Triangle

cC

bB

aA )sin()sin()sin(

==

€

c2 = a2 + b2 − 2abcos(C)

CabA sin21

=

General Sinusoidal Equations

€

y = C + AcosB(x + D)

€

y = C + AsinB(x + D)

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Level 1 Advanced Mathematics Junior Final Exam 2009 Part …...2009 Level 1 Advanced Math Junior Final Exam (Calculator Section) Page 2 Desmond, King, Nitsch, Normile 1. (5 points)