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A new car is purchased for 24900 dollars. The value of the car depreciates at
13.75% per year. What will the value of the car be, to the nearest cent, after 14 years?
Answer:

A new car is purchased for $24900$ dollars. The value of the car depreciates at $13.75 \%$ 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

24900

dollars. The value of the car depreciates at

13.75 \%

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

14

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 $\$24,900$ , and the annual depreciation rate, $r$ , is $13.75\%$ .
Convert Depreciation Rate to Decimal: Convert the annual depreciation rate from a percentage to a decimal. $\newline$ To convert $13.75\%$ to a decimal, divide by $100$ . $\newline$ $r = \frac{13.75\%}{100} = 0.1375$
Determine Depreciation Formula: Determine the formula for depreciation. $\newline$ The value of the car after $t$ years can be calculated using the formula $V(t) = P(1 - r)^t$ , 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 known values into the formula. $\newline$ $P = \$24,900$ , $r = 0.1375$ , and $t = 14$ years. $\newline$ $V(14) = \$24,900(1 - 0.1375)^{14}$
Calculate Value Inside Parentheses: Calculate the value inside the parentheses. $\newline$ $1 - 0.1375 = 0.8625$
Find Depreciation Multiplier: Raise $0.8625$ to the $14$ th power to find the depreciation multiplier. $\newline$ $0.8625^{14}$ is calculated using a calculator.
Perform Calculation: Perform the calculation. $0.8625^{14} \approx 0.2295$ (rounded to four decimal places for intermediate steps)
Multiply Initial Value by Multiplier: Multiply the initial value of the car by the depreciation multiplier. $\$24,900 \times 0.2295 \approx \$5,715.55$
Round Final Value: Round the final value to the nearest cent. $\newline$ The value of the car after $14$ years, to the nearest cent, is approximately $\$5,715.55$ .

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