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?

Explanation:

Step1: Recall the depreciation formula

The formula for exponential depreciation is $V = P(1 - r)^t$, where $V$ is the final value, $P$ is the initial principal (purchase price), $r$ is the annual depreciation rate (as a decimal), and $t$ is the time in years.

Step2: Identify the values

Here, $P = 18000$ dollars, $r = 0.135$ (since $13.5\% = 0.135$), and $t = 14$ years.

Step3: Substitute the values into the formula

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

Step4: Calculate $(1 - 0.135)$

First, calculate $1 - 0.135 = 0.865$.

Step5: Calculate $0.865^{14}$

Using a calculator, $0.865^{14}\approx0.1443$.

Step6: Calculate the final value

Multiply $18000$ by $0.1443$:
$$V = 18000\times0.1443 = 2597.4$$

Answer:

The value of the car after 14 years will be $\$2597.40$ (to the nearest cent).