Question

Salim purchased a new car for $21,000, and it depreciates in value each year. The function y = 21,000(1 − 0.15)x
represents the depreciation. He also purchases a motorcycle, which depreciates each year as well. Its depreciation is represented by the function y = 14,000(1 − 0.09)x . Graph these two functions using Desmos. Which statement best describes when the motorcycle will be worth $7,000?

Statement #1: The motorcycle will have a value of $7,000 after roughly 7.3 years.

Statement #2: The motorcycle will have a value of $7,000 after roughly 6.7 years.

Answer 1

To determine when the motorcycle will be worth $7,000, you can set the depreciation function for the motorcycle equal to $7,000 and solve for \(x\):

The function for the motorcycle is given by:
\[ y = 14,000(1 - 0.09)^x \]

Set \(y\) equal to $7,000:
\[ 14,000(1 - 0.09)^x = 7,000 \]

Now, simplify and solve for \(x\):
\[ (1 - 0.09)^x = \frac{7,000}{14,000} \]
\[ (0.91)^x = 0.5 \]

Next, take the logarithm of both sides:
\[ \log((0.91)^x) = \log(0.5) \]

Using the property of logarithms:
\[ x \cdot \log(0.91) = \log(0.5) \]
\[ x = \frac{\log(0.5)}{\log(0.91)} \]

Now, calculate the value:
- \(\log(0.5) \approx -0.3010\)
- \(\log(0.91) \approx -0.0434\)

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

This value is approximately 6.9 years, which is close to 6.7 years but not quite equal.

Given the options:
- Statement #1: The motorcycle will have a value of $7,000 after roughly 7.3 years.
- Statement #2: The motorcycle will have a value of $7,000 after roughly 6.7 years.

The closest statement to our calculation is **Statement #2**, which suggests that the motorcycle will be worth $7,000 after roughly 6.7 years.