Question

Salim purchased a motorcycle, which depreciates in value each year. It's value is represented by the function y=14,000(1−0.09)^x
. Graph the function using Desmos or GeoGebra. Approximately after how many years will the motorcycle will be worth $7,000 ? (1 point)
Responses

7.3 years
7.3 years

5.6 years

Answer 1

To find out after how many years the motorcycle will be worth $7,000, you can set up the equation based on the depreciation formula:

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

Now, simplify the equation:

1. Divide both sides by 14,000:
\[
\frac{7000}{14000} = (1 - 0.09)^x
\]
\[
0.5 = (0.91)^x
\]

2. Now, take the logarithm of both sides. You can use natural logs (ln) or common logs (log). Here, we'll use natural logs:
\[
\ln(0.5) = \ln(0.91^x)
\]

3. Using the power rule of logarithms, bring down the exponent \(x\):
\[
\ln(0.5) = x \cdot \ln(0.91)
\]

4. Solve for \(x\):
\[
x = \frac{\ln(0.5)}{\ln(0.91)}
\]

Now, calculate the values:

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

So:

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

Thus, the motorcycle will be worth approximately $7,000 after **7.3 years**.

So the correct response is:

**7.3 years**.