MATH1111: College Algebra (20744)
Section 4.1 Exponential Functions
4_1 Homework
4. 1 Homework

Score: $42 / 100 \quad 6 / 15$ answered
Question 13
A vehicle purchased for $\$ 27,500$ depreciates at a constant rate of $8 \%$. Determine the approximate value of the vehicle 12 years after purchase. Round to the nearest whole dollar.
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Step 1 ：Given that the initial value of the vehicle is $27,500, the rate of depreciation is 8% per year, and the time is 12 years. We can use the formula for exponential decay to find the value of the vehicle after 12 years.

Step 2 ：The formula for exponential decay is \(A = P(1 - r)^t\), where \(A\) is the amount of the item after \(t\) years, \(P\) is the principal amount (the initial amount), \(r\) is the rate of depreciation, and \(t\) is the time (in years).

Step 3 ：Substitute \(P = $27,500\), \(r = 8% = 0.08\), and \(t = 12\) years into the formula: \(A = $27,500(1 - 0.08)^{12}\).

Step 4 ：Simplify the equation to get: \(A = $27,500 * (0.92)^{12}\).

Step 5 ：Calculate the value to get: \(A = $27,500 * 0.315020002\).

Step 6 ：Finally, calculate the approximate value of the vehicle 12 years after purchase to get \(\boxed{A = $8,663}\).