Question

Salim 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?

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(1 point)
Responses

6.7 years
6.7 years

7 years
7 years

7.3 years
7.3 years

8 years

Answer 1

To determine when the motorcycle will be worth $7000, we will set the depreciation function equal to $7000:

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

This simplifies to:

\[
14000(0.91)^x = 7000
\]

Next, we can divide both sides by 14000:

\[
(0.91)^x = \frac{7000}{14000} = 0.5
\]

Now, we will take the logarithm of both sides to solve for \(x\):

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

Using the power rule of logarithms:

\[
x \log(0.91) = \log(0.5)
\]

Now, solving for \(x\):

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

Using a calculator to find the values of the logarithms:

\[
\log(0.5) \approx -0.3010
\]
\[
\log(0.91) \approx -0.0434
\]

Now we can substitute these values into the equation:

\[
x = \frac{-0.3010}{-0.0434} \approx 6.93
\]

Rounding to one decimal place, we find that \(x \approx 6.9\) years.

Given the options, the motorcycle will be worth $7000 in approximately **7 years**.