Question

Devon purchased a new car valued at $16,000 that depreciated continuously at a rate of 35%. Its current value is $2,000. The equation 2,000=16,000(1-r)^t represents the situation, where t is the age of the car in years and r is the rate of depreciation. About how old is Devon’s car? Use a calculator and round your answer to the nearest whole number.

1 year
2 years
5 years
8 years

Answer 1

C

Explanation:

Correct on EDGE 2022

Answer 2

9514 1404 393

Answer:

(c) 5 years

Explanation:

A graphing calculator shows the function ...

f(t) = 16000(1 -0.35)^t -2000

will be zero for t ≈ 4.83 years. Rounded to the nearest year, the value is expected to be about $2000 after 5 years.

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Using a scientific calculator, you can rewrite the equation to make use of logarithms.

2000 = 16000(1 -0.35)^t

2000/16000 = 0.65^t

log(1/8) = t·log(0.65)

t = log(0.125)/log(0.65) ≈ 4.827

Devon's car will be about 5 years old when its value is $2000.

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Additional comment

We expected to see the same sort of formula for continuous depreciation that we see for continuous growth: 2000 = 16000e^(-rt). The value formula given in the problem statement is a continuous function, but it should not be described as modeling continuous depreciation.