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Question 1129690: At the beginning of 2010 Dana's car was worth $13,000, but the value of her car decreases exponentially. She notices that the value of her car decreases by 19% every 3 years.
A)What is the 3-year growth factor for the value of Dana's car?
B)What is the 1-year growth factor for the value of Dana's car?
C)Write a function f that determines the value of Dana's car (in dollars) in terms of the number of years t since the beginning of 2010.
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39630)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the formula to use is f = p * (1 + r) ^ t
f is the future value
p is the present value
r is the growth rate per year
t is the number of years
in 3 years, the car reduces 19% in value.
therefore, in 3 years, the value of the car is 13000 minus .19 * 3000 = 10530.
your frmula of f = p * (1 + r) ^ t becomes:
10530 = 13000 * (1 + r) ^ 3
divide both sides of this equation by 1000 to get:
10530 / 13000 = 1 + r) ^ 3
take the third root of both sides of this equation to get:
(10530 / 13000) ^ (1/3) = 1 + r
subtract 1 from both sides of this equation to get:
(10530 / 13000) ^ (1/3) - 1 = r
solve for r to get:
r = -.0678302482
that's your annual growth rate.
your formula becomes:
f = 13000 * (1 - .0678302482) ^ t
in 1 year, the value of the car is 13000 * (1 - .0678302482) ^ 1 = 12118.20677.
in 2 years, the value of the car is 13000 * (1 - .0678302482) ^ 2 = 11296.2258.
in 3 years, the value of the car is 13000 * (1 - .0678302482) ^ 3 = 10530.
the 3 year growth factor is (1 - .0678302482) ^ 3 = .81
the 1 year growth factor is (1 - .0678302482) ^ 1 = .9321697518
the formula to find the value of the car after t years is f = 13000 * (1 - .0678302482) ^ t
that formula can be graphed as shown below.
the formula to graph is y = 13000 * (1 - .0678302482) ^ x.
in the graph, y represents f and x represents t.
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