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A new car is purchased for 19400 dollars. The value of the car depreciates at
13.5% per year. What will the value of the car be, to the nearest cent, after 12 years?
Answer:

A new car is purchased for $19400$ dollars. The value of the car depreciates at $13.5 \%$ per year. What will the value of the car be, to the nearest cent, after $12$ years? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. A new car is purchased for

19400

dollars. The value of the car depreciates at

13.5 \%

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

12

years?

\newline

Answer:

Identify Value and Rate: Identify the initial value of the car and the annual depreciation rate. $\newline$ The initial value of the car, $P$ , is $\$19,400$ , and the annual depreciation rate, $r$ , is $13.5\%$ .
Convert to Decimal: Convert the annual depreciation rate from a percentage to a decimal. $\newline$ To convert a percentage to a decimal, divide by $100$ . $\newline$ $r = \frac{13.5\%}{100} = 0.135$
Determine Depreciation Factor: Determine the depreciation factor. $\newline$ The depreciation factor is what you multiply the car's value by each year to find the next year's value. Since the car is losing value, the factor is $1$ minus the depreciation rate. $\newline$ Depreciation factor = $1 - r = 1 - 0.135 = 0.865$
Apply Exponential Decay Formula: Apply the exponential decay formula to find the car's value after $12$ 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 depreciation rate, and $t$ is the time in years. $\newline$ $V = 19400 \times 0.865^{12}$
Calculate Value After $12$ Years: Calculate the car's value after $12$ years. $\newline$ Using a calculator, raise $0.865$ to the $12^{\text{th}}$ power and then multiply by $19,400$ . $\newline$ $V = 19400 \times 0.865^{12}$ $\newline$ $V \approx 19400 \times 0.1927$ $\newline$ $V \approx 3738.38$
Round Final Value: Round the final value to the nearest cent. $\newline$ The value of the car after $12$ years, rounded to the nearest cent, is approximately $\$3,738.38$ .

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