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.

(1 point)
Statement #

Answer 1

To determine when the motorcycle will be worth $7,000, we can analyze the depreciation function for the motorcycle:

The value of the motorcycle after \( x \) years is given by:

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

We set \( y \) equal to $7,000 to find \( x \):

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

Dividing both sides by 14,000 gives:

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

This simplifies to:

\[ 0.5 = (0.91)^x \]

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

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

Now, we can find \( x \):

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

Calculating \( x \):

1. \(\log(0.5) \approx -0.3010\)
2. \(\log(0.91) \approx -0.0434\)

Thus:

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

Rounding to one decimal place gives:

\[ x \approx 6.9 \]

This indicates that the motorcycle will depreciate to $7,000 in approximately 6.9 years.

Now, reviewing the original 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 close to 6.9 years, which is nearer to Statement #2, we can conclude that:

**Statement #2** is the better choice.