Question

select the correct answer.
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?
a. 6 years
b. 2 years
c. 7 years
d. 4 years

Explanation:

Step1: Set up the depreciation formula

The formula for exponential depreciation is \(A = P(1 - r)^t\), where \(A\) is the final amount, \(P\) is the initial amount, \(r\) is the rate of depreciation, and \(t\) is the time in years. Here, \(P=\$1875\), \(r = 0.2\), and \(A=\$768\). So the equation becomes \(768=1875(1 - 0.2)^t\).

Step2: Simplify the equation

First, divide both sides of the equation by \(1875\): \(\frac{768}{1875}=(0.8)^t\). Then, \(\frac{768}{1875}=\frac{768\div3}{1875\div3}=\frac{256}{625}\). So, \(\frac{256}{625}=(0.8)^t\). Since \(0.8=\frac{4}{5}\) and \(\frac{256}{625}=(\frac{4}{5})^4\), we can see that \(t = 4\).

Answer:

D. 4 years