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 based on its depreciation function, we can set the equation for the motorcycle's value equal to $7,000 and solve for \( x \):

The motorcycle's value function is given by:

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

Setting \( y \) to $7,000:

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

Dividing both sides by 14,000:

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

This simplifies to:

\[ 0.5 = (0.91)^x \]

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

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

Now we solve for \( x \):

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

Using a calculator or logarithm table, find the values of \( \log(0.5) \) and \( \log(0.91) \):

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

Now substituting these into the equation:

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

This means that the motorcycle will have a value of $7,000 after roughly 6.93 years.

Given the provided statements:

- 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.

Since the calculated value is approximately 6.93, **neither statement is exactly correct, but Statement #2 ("roughly 6.7 years") is closer to the calculated value**. Thus, if you have to choose the best option between the two, **you should select Statement #2**.