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A new car is purchased for 18400 dollars. The value of the car depreciates at
10.75% per year. What will the value of the car be, to the nearest cent, after 9 years?
Answer:

A new car is purchased for $18400$ dollars. The value of the car depreciates at $10.75 \%$ per year. What will the value of the car be, to the nearest cent, after $9$ years? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. A new car is purchased for

18400

dollars. The value of the car depreciates at

10.75 \%

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

9

years?

\newline

Answer:

Determine initial value and rate: Determine the initial value of the car and the annual depreciation rate. $\newline$ The initial value of the car, $P$ , is $\$18,400$ , and the annual depreciation rate, $r$ , is $10.75\%$ .
Convert rate to decimal: Convert the annual depreciation rate from a percentage to a decimal. $\newline$ To convert a percentage to a decimal, divide by $100$ . $\newline$ $r = 10.75\% = \frac{10.75}{100} = 0.1075$
Use exponential decay formula: Use the formula for exponential decay to calculate the value of the car after $t$ years. $\newline$ The formula for exponential decay is $V(t) = P(1 - r)^t$ , where $V(t)$ is the value after $t$ years, $P$ is the initial value, and $r$ is the depreciation rate.
Substitute values into formula: Substitute the given values into the formula. $\newline$ $P = \$18,400$ , $r = 0.1075$ , and $t = 9$ years. $\newline$ $V(9) = 18400(1 - 0.1075)^9$
Calculate value inside parentheses: Calculate the value inside the parentheses. $\newline$ $1 - 0.1075 = 0.8925$
Find depreciation factor: Raise $0.8925$ to the $9th$ power to find the depreciation factor. $\newline$ $0.8925^9$ (This calculation can be done using a calculator.)
Calculate value after $9$ years: Calculate the value of the car after $9$ years. $\newline$ $V(9) = 18400 \times 0.8925^9$ $\newline$ Using a calculator, $0.8925^9 \approx 0.424097$ $\newline$ $V(9) = 18400 \times 0.424097$
Perform final multiplication: Perform the multiplication to find the final value. $\newline$ $V(9) \approx 18400 \times 0.424097$ $\newline$ $V(9) \approx 7803.3856$ $\newline$ Round to the nearest cent. $\newline$ $V(9) \approx \$(7,803.39)$

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