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

Step2: Simplify the equation

First, simplify \((1 - 0.2)\) to \(0.8\). The equation is \(768 = 1875\times0.8^t\). Then, divide both sides by \(1875\): \(\frac{768}{1875}=0.8^t\). Calculate \(\frac{768}{1875}=\frac{768\div3}{1875\div3}=\frac{256}{625}\).

Step3: Rewrite the fractions as powers

We know that \(0.8=\frac{4}{5}\), \(\frac{256}{625}=(\frac{4}{5})^4\). So the equation \( \frac{256}{625}=0.8^t\) can be rewritten as \((\frac{4}{5})^4 = (\frac{4}{5})^t\).

Answer:

D. 4 years