Question

alim purchased a motorcycle and it depreciates each year. Its depreciation is represented by the function f(x) = 14000(1−0.09)x When will the motorcycle be worth $7000?

Answer 1

To find out when the motorcycle will be worth $7000, we need to set the given depreciation function equal to $7000 and solve for \( x \):

\[
f(x) = 14000(1 - 0.09)^x
\]

Setting this equal to $7000:

\[
14000(1 - 0.09)^x = 7000
\]

Next, divide both sides by 14000:

\[
(1 - 0.09)^x = \frac{7000}{14000}
\]

This simplifies to:

\[
(1 - 0.09)^x = 0.5
\]

Substituting \( 1 - 0.09 \) with \( 0.91 \):

\[
0.91^x = 0.5
\]

To solve for \( x \), we can take the logarithm of both sides. Using the natural logarithm:

\[
\ln(0.91^x) = \ln(0.5)
\]

Using the power rule of logarithms:

\[
x \cdot \ln(0.91) = \ln(0.5)
\]

Now, solve for \( x \):

\[
x = \frac{\ln(0.5)}{\ln(0.91)}
\]

Calculating the values:

- \( \ln(0.5) \approx -0.6931 \)
- \( \ln(0.91) \approx -0.0943 \)

So,

\[
x \approx \frac{-0.6931}{-0.0943} \approx 7.35
\]

Since \( x \) represents years, we round to the nearest whole number. This means the motorcycle will be worth approximately $7000 after about **7 years**.