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A new car is purchased for
$50,000 and over time its value depreciates by one half every 3.5 years. What is the value of the car 22 years after it was purchased, to the nearest hundred dollars?
Answer:

A new car is purchased for $\$ 50,000$ and over time its value depreciates by one half every $3$ . $5$ years. What is the value of the car $22$ 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

\$ 50,000

and over time its value depreciates by one half every

3

.

5

years. What is the value of the car

22

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

\newline

Answer:

Identify Depreciation Type: Identify the type of depreciation. $\newline$ The car's value decreases by half every $3.5$ years, which indicates exponential decay.
Determine Initial Value and Factor: Determine the initial value $a$ and the decay factor $b$ . The initial value of the car is $\$50,000$ , so $a = 50000$ . The decay factor is one half, so $b = \frac{1}{2}$ .
Calculate Halving Times: Calculate the number of times the car's value will halve. The car's value halves every $3.5$ years, so we divide the total time by the halving period. $t = 22$ years and $T = 3.5$ years. $x = \frac{t}{T} = \frac{22}{3.5} = 6.2857$ Since the value can only halve a whole number of times, we round down to $6$ times.
Use Exponential Decay Formula: Use the exponential decay formula to find the car's value after $22$ years. $\newline$ Exponential Decay: $V(x) = a(b)^{(x)}$ $\newline$ Substitute $50000$ for $a$ , $\frac{1}{2}$ for $b$ , and $6$ for $x$ . $\newline$ $V(x) = 50000\left(\frac{1}{2}\right)^{(6)}$
Evaluate Expression: Evaluate the expression to find the car's value. $\newline$ $V(x) = 50000(\frac{1}{2})^{6}$ $\newline$ $= 50000 \times (\frac{1}{64})$ $\newline$ $= \frac{50000}{64}$ $\newline$ $= 781.25$
Round to Nearest Hundred: Round the value to the nearest hundred dollars. $\newline$ The value of the car is $\$781.25$ , which rounds to $\$800$ when rounded to the nearest hundred dollars.

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