Question 1124504: At the beginning of 2010 Monique's car was worth $10,000, but the value of her car decreases exponentially. She notices that the value of her car decreases by 20% every 3 years.
a. What is the 3-year growth factor for the value of Monique's car?
b. What is the 1-year growth factor for the value of Monique's car?
c. Write a function f that determines the value of Monique's car (in dollars) in terms of the number of years t since the beginning of 2010.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods.
that's the general formula for the future value of something when compound interest is concerned.
compound interest is a form of exponential growth or decay.
if the car cost 10,000 and decreases 20% in value every 3 years, then:
you can solve as follows:
time period is every 3 years.
interest rate is interest rate per every 3 years.
f = p * (1 + r) ^ n becomes f = 10,000 * (1 - .20) ^ n
simplify this to get f = 10,000 * .80 ^ n
after 3 years, the value of the car is 10,000 * .8 ^ 1 = 8000.
after 6 years, the value of the car is 10,000 * .8 ^ 2 = 6400.
after 9 years, the value of the car is 10,000 * .8 ^ 3 = 5121.536
the average annual growth rate would be calculated as follows:
find the future value after one 3 year period.
assuming the present value is one, the formula would become.
f = 1 * (1 - .2) ^ 1 = 1 * .8 = .8
so, the present value becomes .8 after 3 years.
to find the annual growth rate, you would do the following.
f = p * (1 + r) ^ n becomes .8 = 1 * (1 + r) ^ 3
divide both sides of the equation by 1 to get .8 = (1 + r) ^ 3
take the cube root of both sides of the eqution to get .8 ^ (1/3) = 1 + r
subtract 1 from both sides of the equation to get .8 ^ (1/3) - 1 = r
solve for r to get r = -.0716822333.
that's the annual growth rate.
to confirm, use this rate in the original equation of .8 = 1 * (1 + r) ^ 3
you will get .8 = 1 * (1 - .0716822333) ^ 3
simplify to get .8 = .8
if you graph this equation, it will look like this.
from the graph, you can see that the loss in value is 20% every 3 years.
in year 0, the value is 10,000
in year 3, the value is 8000
in year 6, the value is 6400
in year 9, the value is 5120
the every 3 year growth rate is (1 - .2) = .8
the every year growth rate is (1 - .0716822333) = .9283177667
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