Question

exponential functions - basic
question
a new car is purchased for 15300 dollars. the value of the car depreciates at 14.25% per year. what will the value of the car be, to the nearest cent, after 6 years?
answer
attempt 1 out of 2

Explanation:

Step1: Identify the formula for exponential depreciation

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

Step2: Convert the depreciation rate to decimal

The depreciation rate is $14.25\%$, so in decimal form, $r = \frac{14.25}{100} = 0.1425$.

Step3: Identify the values of \(P\), \(r\), and \(t\)

We have $P = 15300$ dollars, $r = 0.1425$, and $t = 6$ years.

Step4: Substitute the values into the formula

\[

$$\begin{align*} V&=15300\times(1 - 0.1425)^6\\ &=15300\times(0.8575)^6 \end{align*}$$

\]

Step5: Calculate \((0.8575)^6\)

First, calculate \((0.8575)^6\approx0.8575\times0.8575\times0.8575\times0.8575\times0.8575\times0.8575\approx0.3969\) (using a calculator for more precise calculation: \((0.8575)^6\approx e^{6\ln(0.8575)}\approx e^{6\times(-0.153)}\approx e^{-0.918}\approx0.3969\))

Step6: Calculate the final value \(V\)

\[
V = 15300\times0.3969\approx15300\times0.3969 = 6072.57
\]

Answer:

\(6072.57\)