A new car is purchased for $20$ , $000$ dollars. The value of the car depreciates at a rate of $3.7 \%$ per year. Which equation represents the value of the car after $3$ years? $\newline$ $V=20,000(1-0.037)$ $\newline$ $V=20,000(0.63)^{3}$ $\newline$ $V=20,000(0.963)(0.963)(0.963)$ $\newline$ $V=20,000(1.037)^{3}$

Full solution

Q. A new car is purchased for

20

000

dollars. The value of the car depreciates at a rate of

3.7 \%

per year. Which equation represents the value of the car after

3

years?

\newline

V=20,000(1-0.037)

\newline

V=20,000(0.63)^{3}

\newline

V=20,000(0.963)(0.963)(0.963)

\newline

V=20,000(1.037)^{3}

Identify values: Identify the initial value, depreciation rate, and time period. $\newline$ Initial value $P$ = $\$20,000$ $\newline$ Depreciation rate $r$ = $3.7\%$ per year $\newline$ Time $t$ = $3$ years $\newline$ We need to find the value of the car after $3$ years, taking into account the annual depreciation rate.
Convert to decimal: Convert the percentage depreciation rate to a decimal. $\newline$ To convert a percentage to a decimal, divide by $100$ . $\newline$ Depreciation rate $(r) = 3.7\% = \frac{3.7}{100} = 0.037$
Determine formula: Determine the formula for depreciation. $\newline$ The value of the car after a certain number of years can be calculated using the formula: $\newline$ $V = P(1 - r)^t$ $\newline$ 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 values into the formula. $\newline$ Substitute $P = \$20,000$ , $r = 0.037$ , and $t = 3$ into the formula. $\newline$ $V = 20,000(1 - 0.037)^3$
Simplify expression: Simplify the expression inside the parentheses. $\newline$ $1 - 0.037 = 0.963$ $\newline$ So the equation becomes: $\newline$ $V = 20,000(0.963)^3$
Check options: Check the given options to see which one matches the simplified equation. $\newline$ The correct equation that represents the value of the car after $3$ years is: $\newline$ $V = 20,000(0.963)^3$ $\newline$ This matches one of the given options.