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A new car is purchased for 16900 dollars. The value of the car depreciates at
15% per year. What will the value of the car be, to the nearest cent, after 15 years?
Answer:

A new car is purchased for $16900$ dollars. The value of the car depreciates at $15 \%$ per year. What will the value of the car be, to the nearest cent, after $15$ years? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. A new car is purchased for

16900

dollars. The value of the car depreciates at

15 \%

per year. What will the value of the car be, to the nearest cent, after

15

years?

\newline

Answer:

Identify Value and Rate: Identify the initial value of the car and the rate of depreciation. The initial value of the car, $P$ , is \$\(16\),\(900\), and the annual rate of depreciation is \(15\%\).
Convert to Decimal: Convert the percentage of depreciation to a decimal.\(\newline\)To convert a percentage to a decimal, divide by \(100\).\(\newline\)Depreciation rate in decimal form = \(\frac{15\%}{100} = 0.15\)
Determine Depreciation Factor: Determine the depreciation factor.\(\newline\)The depreciation factor is what the value is multiplied by each year to account for the loss in value. Since the car loses \(15\%\) of its value each year, it retains \(85\%\) of its value.\(\newline\)Depreciation factor = \(1 - \) depreciation rate = \(1 - 0.15 = 0.85\)
Apply Exponential Decay Formula: Apply the exponential decay formula to find the value of the car after \(15\) years.\(\newline\)The formula for exponential decay 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.\(\newline\)\(V = 16900 \times (0.85)^{15}\)
Calculate Final Value: Calculate the value of the car after \(15\) years.\(\newline\)\(V = 16900 \times (0.85)^{15}\)\(\newline\)Using a calculator, we find:\(\newline\)\(V \approx 16900 \times 0.087\)\(\newline\)\(V \approx 1470.30\)

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