Question

a new car is purchased for 15500 dollars. the value of the car depreciates at 7.5% per year. what will the value of the car be, to the nearest cent, after 6 years?

Explanation:

Step1: Identify the formula for depreciation

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

Step2: Convert the rate to a decimal

The rate of depreciation is $7.5\%$, so converting to a decimal gives $r = 0.075$.

Step3: Substitute the values into the formula

We have $P = 15500$, $r = 0.075$, and $t = 6$. Substituting into the formula $V = P(1 - r)^t$ gives:

$$ V = 15500(1 - 0.075)^6 $$

Step4: Calculate $(1 - 0.075)$

First, calculate $1 - 0.075 = 0.925$.

Step5: Calculate $0.925^6$

Using a calculator, $0.925^6 \approx 0.6328$.

Step6: Multiply by the initial value

Now, multiply this by the initial value $15500$:

$$ V = 15500 \times 0.6328 \approx 9808.4 $$

Answer:

The value of the car after 6 years will be approximately $\$9808.40$.