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Calculus IChapter 1: ReviewSection 1-7

Exponential Functions — Question 4

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Question 4

The value of a car decreases exponentially over time according to the function V(t)=25000e−0.18tV(t) = 25000e^{-0.18t} where V(t)V(t) is the value in dollars and tt is the number of years since purchase.

  • (a) What is the value of the car after 3 years?

  • (b) How long will it take for the car to be worth half its original value?

  • (c) After how many years will the car be worth $5,000?

Original worksheet page 1: question and worked solution for 1-7-004
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Question 4 - Solution

Use the exact exponential model and keep full precision until the final rounding.

(a)

V(3)=25000e−0.54V(3)=25000e^{-0.54}

14568.71dollars\boxed{14568.71\quad\text{dollars}}

(b)

e−0.18t=1/2⇒t=ln⁡(2)/0.18e^{-0.18t}=1/2\ \Longrightarrow\ t=\ln(2)/0.18

3.85years\boxed{3.85\quad\text{years}}

(c)

25000e−0.18t=5000⇒t=ln⁡(5)/0.1825000e^{-0.18t}=5000\ \Longrightarrow\ t=\ln(5)/0.18

8.94years\boxed{8.94\quad\text{years}}

Original worksheet page 2: question and worked solution for 1-7-004

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