Question

Cameron purchased an electric guitar for $1,875. The value of the guitar depreciates by 20% each year. In how many years will the guitar be valued at $768?

Answer 1

To determine in how many years the value of Cameron's guitar will depreciate to $768, we can use the formula for exponential decay. The value of the guitar after \( t \) years can be represented by the equation:

\[
V = P(1 - r)^t
\]

Where:
- \( V \) is the future value of the guitar,
- \( P \) is the initial purchase price of the guitar,
- \( r \) is the depreciation rate (as a decimal),
- \( t \) is the number of years.

Given:
- \( P = 1875 \)
- \( r = 0.20 \) (which is 20% as a decimal)
- \( V = 768 \)

We substitute the known values into the formula:

\[
768 = 1875(1 - 0.20)^t
\]
\[
768 = 1875(0.80)^t
\]

Next, we isolate \( (0.80)^t \):

\[
(0.80)^t = \frac{768}{1875}
\]

Calculating the right side:

\[
\frac{768}{1875} \approx 0.4096
\]

Now we have:

\[
(0.80)^t \approx 0.4096
\]

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

\[
\log((0.80)^t) = \log(0.4096)
\]

Using the logarithmic power rule:

\[
t \log(0.80) = \log(0.4096)
\]

Now, solving for \( t \):

\[
t = \frac{\log(0.4096)}{\log(0.80)}
\]

Calculating the logarithms:

Using a calculator to find values:
- \( \log(0.4096) \approx -0.3860 \)
- \( \log(0.80) \approx -0.0969 \)

Now substituting these values back to find \( t \):

\[
t \approx \frac{-0.3860}{-0.0969} \approx 3.98
\]

Since \( t \) must be a whole number, we round up to the next whole number:

Thus, it will take approximately **4 years** for the guitar to depreciate to about $768.

Therefore, the answer is:

\[
\boxed{4}
\]