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A new car is purchased for 19600 dollars. The value of the car depreciates at
11.75% per year. What will the value of the car be, to the nearest cent, after 14 years?
Answer:

A new car is purchased for $19600$ dollars. The value of the car depreciates at $11.75 \%$ per year. What will the value of the car be, to the nearest cent, after $14$ years? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. A new car is purchased for

19600

dollars. The value of the car depreciates at

11.75 \%

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

14

years?

\newline

Answer:

Identify values: Identify the initial value of the car and the annual depreciation rate. $\newline$ The initial value of the car, $P$ , is $\$19,600$ , and the annual depreciation rate, $r$ , is $11.75\%$ .
Convert to decimal: Convert the annual depreciation rate from a percentage to a decimal. $\newline$ To convert the rate to a decimal, divide by $100$ . $\newline$ $r = \frac{11.75\%}{100} = 0.1175$
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 = $1 - r = 1 - 0.1175 = 0.8825$
Apply factor over $14$ years: Apply the depreciation factor over the course of $14$ years. $\newline$ To find the value of the car after $14$ years, we raise the depreciation factor to the power of $14$ and multiply it by the initial value. $\newline$ Value after $14$ years = $P \times (\text{Depreciation factor})^{14}$
Calculate value after $14$ years: Calculate the value of the car after $14$ years. $\newline$ Value after $14$ years = $\$19,600 \times 0.8825^{14}$
Perform calculation: Perform the calculation. $\newline$ Using a calculator, we find: $\newline$ Value after $14$ years = $\$19,600 \times 0.8825^{14} \approx \$19,600 \times 0.2261 \approx \$4,431.76$
Round final value: Round the final value to the nearest cent. $\newline$ The value of the car after $14$ years, rounded to the nearest cent, is approximately $\$4431.76$ .

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