Exponential Functions — Question 4

Question 4

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

• What is the value of the car after 3 years?

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

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

Question 4 - Solution

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

(a)

$$V(3)=25000e^{-0.54}$$

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

(b)

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

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

(c)

$$25000e^{-0.18t}=5000\ \Longrightarrow\ t=\ln(5)/0.18$$

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