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A new car is purchased for
$44,000 and over time its value depreciates by one half every 5 years. What is the value of the car 24 years after it was purchased, to the nearest hundred dollars?
Answer:

A new car is purchased for $\$ 44,000$ and over time its value depreciates by one half every $5$ years. What is the value of the car $24$ years after it was purchased, to the nearest hundred dollars? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. A new car is purchased for

\$ 44,000

and over time its value depreciates by one half every

5

years. What is the value of the car

24

years after it was purchased, to the nearest hundred dollars?

\newline

Answer:

Identify Values: Identify the initial value of the car and the rate of depreciation. $\newline$ The initial value of the car is $\$44,000$ , and it depreciates by half every $5$ years.
Determine Intervals: Determine the number of $5$ -year intervals in $24$ years. $\newline$ Since the car depreciates every $5$ years, we divide $24$ by $5$ to find the number of intervals. $\newline$ Number of intervals = $24 / 5 = 4.8$ $\newline$ Since depreciation happens at discrete $5$ -year intervals, we round down to $4$ intervals for the calculation, as the car will not have completed its $5$ th interval of depreciation at the $24$ -year mark.
Calculate Value: Calculate the value of the car after each $5$ -year interval. $\newline$ We use the formula for exponential decay, which is Final Value = Initial Value $\times$ $(1/2)^{\text{number of intervals}}$ . $\newline$ Final Value = $\$44,000 \times (1/2)^{4}$
Perform Calculation: Perform the calculation for the exponential decay. $\newline$ Final Value = $\$44,000 \times \frac{1}{16}$ $\newline$ Final Value = $\$44,000 \times 0.0625$ $\newline$ Final Value = $\$2,750$
Round Final Value: Round the final value to the nearest hundred dollars. $\newline$ The value of the car $24$ years after it was purchased, rounded to the nearest hundred dollars, is $\$2,800$ .

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