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In the year 2006 , a person bought a new car for
$30000. For each consecutive year after that, the value of the car depreciated by
8%. How much would the car be worth in the year 2010, to the nearest hundred dollars?
Answer:

In the year $2006$ , a person bought a new car for $\$ 30000$ . For each consecutive year after that, the value of the car depreciated by $8 \%$ . How much would the car be worth in the year $2010$ , to the nearest hundred dollars? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. In the year

2006

, a person bought a new car for

\$ 30000

. For each consecutive year after that, the value of the car depreciated by

8 \%

. How much would the car be worth in the year

2010

, to the nearest hundred dollars?

\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 $\$30,000$ . The annual depreciation rate, $r$ , is $8\%$ .
Convert rate to decimal: Convert the annual depreciation rate from a percentage to a decimal. $\newline$ To convert $8\%$ to a decimal, divide by $100$ : $r = \frac{8}{100} = 0.08$ .
Calculate years of depreciation: Calculate the number of years, $t$ , the car has depreciated from $2006$ to $2010$ . The number of years is $t = 2010 - 2006 = 4$ years.
Use exponential decay formula: Use the formula for exponential decay to find the value of the car after $t$ years. $\newline$ The formula is $V = P(1 - r)^t$ , where $V$ is the final value.
Substitute values and calculate: Substitute the known values into the formula and calculate the value of the car in $2010$ . $\newline$ $V = 30000(1 - 0.08)^4$ $\newline$ $V = 30000(0.92)^4$
Calculate $(0.92)^4$ : Calculate the value of $(0.92)^4.$ $\newline$ $(0.92)^4 \approx 0.71639296$
Multiply initial value by factor: Multiply the initial value of the car by the depreciation factor to find the final value. $\newline$ $V \approx 30000 \times 0.71639296$ $\newline$ $V \approx 21491.7888$
Round final value: Round the final value to the nearest hundred dollars as requested. $\newline$ The car would be worth approximately $\$21,500$ in the year $2010$ .

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