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Barry just read that his computer, which costs $\$1,300$ new, loses $25\%$ of its value every year. If this estimate is accurate, how much will the computer be worth in $15$ years? If necessary, round your answer to the nearest cent. $\newline$ $\$$ ____

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Exponential growth and decay: word problems

Full solution

Q. Barry just read that his computer, which costs

\$1,300

new, loses

25\%

of its value every year. If this estimate is accurate, how much will the computer be worth in

15

years? If necessary, round your answer to the nearest cent.

\newline

\$

____

Identify initial value and rate: Identify the initial value of the computer and the rate of depreciation. $\newline$ The initial value of the computer is $\$1,300$ , and it depreciates by $25\%$ each year.
Determine depreciation factor: Determine the depreciation factor. $\newline$ Since the computer loses $25\%$ of its value each year, it retains $75\%$ of its value each year. To find the depreciation factor, convert the percentage to a decimal. $\newline$ Depreciation factor = $100\% - 25\% = 75\% = 0.75$
Apply exponential decay formula: Apply the formula for exponential decay to find the value of the computer after $15$ years. $\newline$ The formula for exponential decay is $P(t) = P_0 \times (1 - r)^t$ , where $P(t)$ is the future value, $P_0$ is the initial value, $r$ is the rate of depreciation, and $t$ is the time in years. $\newline$ In this case, $P_0 = \$1,300$ , $r = 0.25$ , and $t = 15$ .
Calculate value after $15$ years: Calculate the value of the computer after $15$ years. $\newline$ $P(15) = \$(1,300) \times (0.75)^{15}$ $\newline$ First, calculate $(0.75)^{15}$ . $\newline$ $(0.75)^{15} \approx 0.013031$ $\newline$ Now, multiply this by the initial value. $\newline$ $P(15) = \$(1,300) \times 0.013031 \approx \$(16.9403)$
Round to nearest cent: Round the answer to the nearest cent. $\newline$ The value of the computer after $15$ years, rounded to the nearest cent, is approximately $\$16.94$ .

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