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In the year 2002, a person bought a new car for
$18500. For each consecutive year after that, the value of the car depreciated by
7%. How much would the car be worth in the year 2004, to the nearest hundred dollars?
Answer:

In the year $2002$ , a person bought a new car for $\$ 18500$ . For each consecutive year after that, the value of the car depreciated by $7 \%$ . How much would the car be worth in the year $2004$ , to the nearest hundred dollars? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. In the year

2002

, a person bought a new car for

\$ 18500

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

7 \%

. How much would the car be worth in the year

2004

, to the nearest hundred dollars?

\newline

Answer:

Identify initial value and rate: Identify the initial value of the car and the annual depreciation rate. $\newline$ The initial value of the car, $P$ , is $\$18500$ , and the annual depreciation rate, $r$ , is $7\%$ .
Convert rate to decimal: Convert the annual depreciation rate from a percentage to a decimal. $\newline$ To convert $7\%$ to a decimal, divide by $100$ : $r = \frac{7}{100} = 0.07$ .
Determine number of years: Determine the number of years, $t$ , the car has depreciated. $\newline$ Since the car was bought in $2002$ and we want to find its value in $2004$ , $t = 2004 - 2002 = 2$ years.
Use depreciation formula: Use the formula for depreciation to calculate the value of the car after $t$ years. $\newline$ The formula for depreciation is $V = P(1 - r)^t$ , where $V$ is the value after $t$ years.
Substitute values and calculate: Substitute the known values into the formula and calculate the value of the car in $2004$ . $\newline$ $V = 18500(1 - 0.07)^2$ $\newline$ $V = 18500(0.93)^2$ $\newline$ $V = 18500 \times 0.8649$ (rounded to four decimal places)
Perform multiplication: Perform the multiplication to find the depreciated value of the car. $\newline$ $V = 18500 \times 0.8649$ $\newline$ $V = 16010.65$
Round to nearest hundred: Round the value to the nearest hundred dollars as requested. $\newline$ The value $\$16010.65$ rounded to the nearest hundred dollars is $\$16000$ .

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