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A new car is purchased for
$44,000 and over time its value depreciates by one half every 5.5 years. What is the value of the car 9 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 $5$ . $5$ years. What is the value of the car $9$ 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

.

5

years. What is the value of the car

9

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

\newline

Answer:

Determine initial value and rate: Determine the initial value of the car and the rate of depreciation. $\newline$ The initial value of the car, $a$ , is $\$44,000$ . The car depreciates to half its value every $5.5$ years, which means the rate of depreciation, $b$ , is $\frac{1}{2}$ or $0.5$ .
Calculate halving periods: Calculate the number of times the car's value will halve in $9$ years. $\newline$ The car's value halves every $5.5$ years, so we divide the total time by the halving period to find the number of times the value halves. $\newline$ $x = \frac{t}{T}$ $\newline$ $t = 9$ (total time in years) and $T = 5.5$ (time for one halving) $\newline$ $x = \frac{t}{T}$ $\newline$ $= \frac{9}{5.5}$
Evaluate number of periods: Evaluate the number of halving periods. $\newline$ $x = \frac{9}{5.5}$ $\newline$ $= 1.63636...$ $\newline$ Since the value of the car can only halve a whole number of times, we need to round down to the nearest whole number. The car's value will halve $1$ time in $9$ years.
Apply depreciation formula: Apply the exponential depreciation formula to find the car's value after $9$ years. $\newline$ Depreciation formula: $V(x) = a(b)^{(x)}$ $\newline$ $V(x) = 44000(0.5)^{(1)}$
Calculate value after halving: Calculate the car's value after it has halved once. $\newline$ $V(x) = 44000(0.5)^{1}$ $\newline$ $= 44000 \times 0.5$ $\newline$ $= 22000$ $\newline$ The car's value after $9$ years is $\$22,000$ before rounding to the nearest hundred dollars.
Round value to nearest hundred: Round the car's value to the nearest hundred dollars. The value of the car is $\$22,000$ , which is already rounded to the nearest hundred dollars.

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