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
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
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.

Answer 1

To solve this problem, we need to find out when the value of the motorcycle, represented by the equation

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

will equal $7,000.

Let’s set the equation equal to 7,000:

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

First, divide both sides by 14,000:

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

This simplifies to:

\[
(1 - 0.09)^x = 0.5
\]

Now we'll compute \(1 - 0.09\):

\[
0.91^x = 0.5
\]

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

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

Using the logarithm power rule, we can simplify:

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

Now, solve for \(x\):

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

Now calculating \(x\):
- \(\log(0.5) \approx -0.3010\)
- \(\log(0.91) \approx -0.0434\)

Thus:

\[
x \approx \frac{-0.3010}{-0.0434} \approx 6.93 \text{ (approximately)}
\]

This implies that it will take around 6.93 years for the motorcycle to be worth $7,000.

Given the 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 6.93 is closer to Statement #2 than #1, the correct answer is:

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