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In the year 2000, a person bought a new car for
$16500. For each consecutive year after that, the value of the car depreciated by
6%. How much would the car be worth in the year 2003 , to the nearest hundred dollars?
Answer:

In the year $2000$ , a person bought a new car for $\$ 16500$ . For each consecutive year after that, the value of the car depreciated by $6 \%$ . How much would the car be worth in the year $2003$ , to the nearest hundred dollars? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. In the year

2000

, a person bought a new car for

\$ 16500

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

6 \%

. How much would the car be worth in the year

2003

, 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 is $\$16,500$ , and it depreciates by $6\%$ each year.
Calculate Depreciation Multiplier: Calculate the depreciation multiplier. $\newline$ The car loses $6\%$ of its value each year, so it retains $94\%$ of its value each year ( $100\% - 6\% = 94\%$ ). $\newline$ To find the multiplier, convert the percentage to a decimal: $94\% = 0.94$ .
Determine Years Depreciated: Determine the number of years the car has depreciated. $\newline$ From the year $2000$ to the year $2003$ is $3$ years.
Apply Depreciation Formula: Apply the depreciation formula to calculate the car's value in $2003$ . $\newline$ The formula for depreciation is: $\text{Final Value} = \text{Initial Value} \times (\text{Depreciation Multiplier})^{(\text{Number of Years})}$ . $\newline$ Substitute the known values into the formula: $\text{Final Value} = \$(16,500) \times (0.94)^3$ .
Perform Calculation: Perform the calculation. $\newline$ Final Value = $\$16,500 \times (0.94)^3$ $\newline$ Final Value = $\$16,500 \times 0.830584$ ( $0.94$ cubed is approximately $0.830584$ ) $\newline$ Final Value = $\$13,711.64$
Round Final Value: Round the final value to the nearest hundred dollars. $\newline$ The value of the car in $2003$ , rounded to the nearest hundred dollars, is approximately $\$13,700$ .

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