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 (in decimal), and $t$ is the time in years.

Step2: Convert the rate to decimal

The rate $r = 7.5\% = 0.075$.

Step3: Substitute the values into the formula

Here, $P = 15500$, $r = 0.075$, and $t = 6$. So we have $V = 15500(1 - 0.075)^6$.

Step4: Calculate the value inside the parentheses

$1 - 0.075 = 0.925$.

Step5: Calculate $(0.925)^6$

First, calculate $0.925^2 = 0.925\times0.925 = 0.855625$. Then $0.925^4=(0.855625)^2 = 0.73210140625$. Then $0.925^6 = 0.925^4\times0.925^2=0.73210140625\times0.855625\approx0.626141$.

Step6: Multiply by the initial value

$V = 15500\times0.626141\approx15500\times0.626141 = 9705.1855$. Rounding to the nearest cent, we get $V\approx9705.19$.

Answer:

The value of the car after 6 years will be approximately $\$9705.19$.