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 GeoGebra. 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.

(1 point)
Statement #

Answer 1

To determine when the motorcycle will be worth $7,000, we need to solve the equation given by the depreciation function for the motorcycle:

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

We need to set \( y \) to $7,000 and solve for \( x \):

\[ 7,000 = 14,000(1 - 0.09)^x \]

First, we can simplify this equation:

\[ \frac{7,000}{14,000} = (1 - 0.09)^x \]

This simplifies to:

\[ 0.5 = (0.91)^x \]

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

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

Now, we isolate \( x \):

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

Using a calculator to find the logarithmic values:

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

Therefore:

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

This means the motorcycle will be worth $7,000 after approximately 6.93 years.

Given the choices:

- 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 best statement that aligns with our finding (6.93 years) is **Statement #2**.

Thus, the answer is **Statement #2**.