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:
\[ \boxed{4} \]
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