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 car will be worth $4,000?



Statement #1: The car will have a value of $4,000 after roughly 10.2 years.

Statement #2: The car will have a value of $4,000 after roughly 13.3 years.

(1 point)
Statement #

Answer 1

To determine when the car will be worth $4,000 using the given depreciation function \( y = 21,000(1 - 0.15)^x \), we can set \( y \) equal to $4,000 and solve for \( x \):

\[
4000 = 21000(1 - 0.15)^x
\]

First, simplify the equation:

\[
\frac{4000}{21000} = (0.85)^x
\]

Calculating the left side gives:

\[
\frac{4000}{21000} \approx 0.1905
\]

Now we have:

\[
0.1905 = (0.85)^x
\]

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

\[
\log(0.1905) = x \log(0.85)
\]

Finally, we can isolate \( x \):

\[
x = \frac{\log(0.1905)}{\log(0.85)}
\]

Using a calculator, we find:

\[
\log(0.1905) \approx -0.7203 \quad \text{and} \quad \log(0.85) \approx -0.0706
\]

So,

\[
x \approx \frac{-0.7203}{-0.0706} \approx 10.2
\]

Thus, the car will be worth $4,000 after approximately 10.2 years.

Therefore, the correct statement is:

**Statement #1: The car will have a value of $4,000 after roughly 10.2 years.**