Exponential Functions Topic 3: Applications of Exponential
Functions
Slide 2
Solve problems that involve the application of exponential
equations to loans, mortgages, and investments. Graph data, and
determine the exponential function that best approximates the
data.
Slide 3
Explore The rate of depreciation of a new vehicle will vary
depending on the make and model of the vehicle. New cars depreciate
about 20% the moment they are driven off of the lot. The accepted
average car depreciation rate is roughly 15% per year after the
first year. Jodi bought a new car one year ago, and today it is
worth $20 000. Assume a constant depreciation rate of 15% /a.
Interest rates are usually expressed as a percent per annum
(/a).
Slide 4
Explore 1. Calculate the amount that Jodis car will depreciate
over the next year. It will depreciate by $3 000 in the next year.
2. Using the amount from question 1, calculate the value of Jodis
car one year from today. It will be worth $17 000 one year from
today.
Slide 5
Explore 3.Explain how to calculate the value from question 2
using only one step. depreciation so 1- 0.15 = 0.85 of the value
remaining 4. Show how to calculate the amount that Jodis car will
be worth a. two years from today b. three years from today
Slide 6
Explore 5.Determine an equation that you can use to calculate
the value of the car x years from today. a is the initial value of
the car b is the common ratio, depreciation so 1- 0.15
Slide 7
Information The general form of exponential equations is y
represents the final amount, a represents the initial amount, b
represents the common ratio, and x represents the input or
independent variable
Slide 8
Information
Slide 9
There are many real-life applications of exponential equations.
Some of the most common are listed: Compound interest is the
interest earned on both the original amount of money that was
invested and any interest that has accumulated over time. It is
sometimes referred to as earning interest on top of interest. When
money earns interest or is charged interest, an exponential
function can be used to model the situation. The formula for
compound interest is where A(n) is the future value, P is the
principal, i is the interest rate per compounding period, and n
represents the number of compounding periods.
Slide 10
Information The interest rate per compounding period, i, is the
interest rate, expressed as a decimal, that is charged or earned
during one compounding period. The compounding period, n, is the
time over which interest is calculated and paid on an investment or
loan. Some common compounding periods are daily, weekly, monthly,
quarterly, semi-annually, and annually. Interest is calculated at
the end of every compounding period and added to the balance. The
new total becomes the starting value for the next period.
Slide 11
Information The half-life exponential function is used to
calculate exponential decay using the half-life (the amount of time
it takes something to decrease to half of the original amount). It
is represented by the function where A(t) represents the final
amount, A o represents the initial amount, t represents the time,
and h represents the half life.
Slide 12
Information The doubling period exponential function is used to
calculate exponential growth using the doubling period (the amount
of time it takes something to double fro its original amount). It
is represented by the function where N(t) represents the final
amount, N o represents the initial amount, t represents the time,
and d represents the doubling period.
Slide 13
Example 1 The number of movie tickets that are sold in Edmonton
on Friday night is increasing at a rate of 5% each week. In week 0,
2500 tickets were sold. The number of tickets sold, N(t), after t
weeks can be represented by an exponential equation. a) How many
tickets, to the nearest whole ticket will be sold in week 12?
Determining and solving an equation in context An increase by 5%
each week means that the constant ratio (b) is 1.05. The initial
population (y-intercept or a-value) is 2500. 12 weeks later, so x =
12. Solve for y.
Slide 14
Example 1 b) In how many weeks, to the nearest tenth, would
Friday night ticket sales reach 5000 tickets? Determining and
solving an equation in context Divide both sides by 2500. There is
no common base, so I must solve graphically! Graph left side in Y1
and right side in Y2. Window X: [0, 20, 2] Y: [0, 1, 0.2] Using 2
nd Trace 5: intersect, x = 14.2 weeks
Slide 15
Example 2 When diving underwater, the light decreases by 10%
for every metre that the diver dives. On a sunny day off the coast
of Vancouver Island, a diving team recorded a 100% visibility at
the surface. a) Write the exponential equation that represent this
situation where V(d) is the visibility at any given depth, d, in
metres. Determining and solving an equation in context Initial
visibility (at 0m below the surface). Visibility decrease by 10%
per metre, so rate of decrease is 1 0.10.
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Example 2 b) At what depth with the visibility be half the
visibility at the surface? Round to the nearest tenth. The
visibility at the depth we are finding is 50% (half of 100%), so
this is our V(d) value. Now we solve for d. Since the two sides
cannot be written with a common base, solve graphically. Enter left
side into Y1 and the right side into Y2. Solve for the x-value of
the intersection point. Depth: 6.6 metres
Slide 17
Example 3 Olga invested $8 000 in an investment for 5 years
compounded annually at a rate of 7.25%/a. a) Algebraically
determine the value of her investment at the end of the term.
Determining the value of an investment that is compounded annually
Interest rates are usually expressed as a percent per annum
(/a).
Slide 18
Example 3 b) Calculate the interest earned on her investment.
Since she started with $8000 and ends up with $11352.11, she has
earned $3352.11 in interest (11352.11 8000).
Slide 19
Example 4 Jarrod invests $5000. The bond he chooses has an
annual appreciation rate of 2.8%/a, compounded annually. a) How
long, to the nearest tenth of a year, will it take for his
investment to double? Determining the initial investment
Appreciation is the increase in the value of an asset over time.
The accumulated amount (A) is 10 000. The interest rate per period
(i) is 0.028 The principal is $5000. Solve for n.
Slide 20
Example 4 Continued Determining the initial investment
Appreciation is the increase in the value of an asset over time.
Since the two sides cannot be written with the same base, solve
graphically. Enter left side into Y1 and the right side into Y2.
Solve for the x-value of the intersection point. His money reaches
a value of $10 000 after 25.1 years.
Slide 21
Example 4 b) Suppose the loan was instead compounded monthly.
i.Determine how long it will now take for the investment to double
in value, to the nearest month. Use the equation where x is the
number of months. Determining the initial investment Appreciation
is the increase in the value of an asset over time. The accumulated
amount (A) is 10 000. The interest rate per period (i) is The
principal is $5000. Solve for x.
Slide 22
Example 4 ii. How long will it take in years, to the nearest
year? Determining the initial investment Appreciation is the
increase in the value of an asset over time. Since the two sides
cannot be written with the same base, solve graphically. Enter left
side into Y1 and the right side into Y2. Solve for the x-value of
the intersection point. His money reaches a value of $10 000 after
297 months. Continued
Slide 23
Example 4 Determining the initial investment Continued c)
Explain how changing a compounding period affects how an investment
will grow. As the compounding period increases the length of time
for the investment to grow decreases.
Slide 24
Example 5 When an animal dies, the amount of radioactive
carbon-14 (C-14) in its bones decreases. Archaeologists use this
fact to determine the age of a fossil based on the amount of C-14
remaining. The half-life of C-14 is 5 730 years. Head-Smashed-In
Buffalo Jump in southwestern Alberta is recognized as the best
example of a buffalo jump in North America. The oldest bones
unearthed at the site had 49.5% of the C-14 left. Use the formula,
where t is the number of years, and y is the remaining percentage
of C-14, to determine the age of the bones when they were found, to
the nearest year. Using half life to solve a contextual
problem
Slide 25
Example 5 Using half life to solve a contextual problem I can
solve graphically! Graph left side in Y1 and right side in Y2.
Window X: [0, 7000, 500] Y: [0, 100, 10] Using 2 nd Trace 5:
intersect, x = 5813 years.
Slide 26
The general form of exponential equations is y represents the
final amount, a represents the initial amount, b represents the
common ratio, and x represents the input or independent variable
Need to Know
Slide 27
Slide 28
Compound interest is the interest earned on both the original
amount of money that was invested and any interest that has
accumulated over time. It is sometimes referred to as earning
interest on top of interest. When money earns or is charged
interest, an exponential function can be used to model the
situation. The formula for compound interest is
Slide 29
Need to Know For,
Slide 30
Need to Know interest earned = final closing balance total
amount invested (commonly referred to as profit) interest paid =
total amount paid amount borrowed The higher the number of
compounding periods, the faster an investment will grow and the
faster the interest will accumulate. Appreciation is the increase
in the value of an asset over time. Depreciation is the decrease in
the value of an asset over time. Youre ready! Try the homework from
this section.
