Question

question
a new car is purchased for 18000 dollars. the value of the car depreciates at 13.5% per year. what will the value of the car be, to the nearest cent, after 14 years?
answer

Explanation:

Step1: Identify the formula for depreciation

The formula for exponential depreciation is $V = P(1 - r)^t$, where:

• $V$ is the value of the asset after $t$ years,
• $P$ is the initial value (purchase price),
• $r$ is the annual depreciation rate (as a decimal),
• $t$ is the number of years.

Step2: Convert the depreciation rate to a decimal

The depreciation rate is $13.5\%$, so $r = \frac{13.5}{100} = 0.135$.

Step3: Identify the values of $P$, $r$, and $t$

• $P = 18000$ dollars,
• $r = 0.135$,
• $t = 14$ years.

Step4: Substitute the values into the formula

Substitute $P = 18000$, $r = 0.135$, and $t = 14$ into the formula $V = P(1 - r)^t$:

$$ V = 18000(1 - 0.135)^{14} $$

First, calculate $(1 - 0.135) = 0.865$. Then, calculate $0.865^{14}$.

Using a calculator, $0.865^{14} \approx 0.1447$.

Then, $V = 18000 \times 0.1447 \approx 2604.60$.

Answer:

The value of the car after 14 years will be approximately $\$2604.60$.