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A new car is purchased for 23400 dollars. The value of the car depreciates at
5.75% per year. What will the value of the car be, to the nearest cent, after 9 years?
Answer:

A new car is purchased for $23400$ dollars. The value of the car depreciates at $5.75 \%$ per year. What will the value of the car be, to the nearest cent, after $9$ years? $\newline$ Answer:

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Exponential growth and decay: word problems

Full solution

Q. A new car is purchased for

23400

dollars. The value of the car depreciates at

5.75 \%

per year. What will the value of the car be, to the nearest cent, after

9

years?

\newline

Answer:

Determine Initial Value and Rate: Determine the initial value of the car and the annual depreciation rate. $\newline$ Initial value $P$ = $\$23,400$ $\newline$ Annual depreciation rate $r$ = $5.75\%$
Convert to Decimal: Convert the annual depreciation rate from a percentage to a decimal. $r = 5.75\% = 0.0575$
Identify Depreciation Formula: Identify the formula for depreciation using the exponential decay model. $\newline$ The formula is $V = P(1 - r)^t$ , where $V$ is the final value, $P$ is the initial value, $r$ is the depreciation rate, and $t$ is the time in years.
Substitute Values: Substitute the given values into the formula to calculate the value of the car after $9$ years. $\newline$ $V = 23400(1 - 0.0575)^9$
Calculate Inside Parentheses: Calculate the value inside the parentheses first. $1 - 0.0575 = 0.9425$
Calculate Depreciation Factor: Raise $0.9425$ to the power of $9$ to find the depreciation factor. $\newline$ $0.9425^9 \approx 0.6057$
Find Final Value: Multiply the initial value of the car by the depreciation factor to find the final value. $\newline$ $V \approx 23400 \times 0.6057 \approx 14173.38$

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