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A piece of land was purchased for
$65,000. The value of the land has slowly been decreasing by
1% annually. How much will the land be worth in 20 years? Round your answer to the nearest dollar.

A piece of land was purchased for $\$ 65,000$ . The value of the land has slowly been decreasing by $1 \%$ annually. How much will the land be worth in $20$ years? Round your answer to the nearest dollar. $\newline$

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Exponential growth and decay: word problems

Full solution

Q. A piece of land was purchased for

\$ 65,000

. The value of the land has slowly been decreasing by

1 \%

annually. How much will the land be worth in

20

years? Round your answer to the nearest dollar.

\newline

Identify Value and Percentage: Identify the initial value of the land and the annual percentage decrease. $\newline$ The initial value of the land, $P_0$ , is $\$65,000$ , and the annual percentage decrease is $1\%$ .
Convert to Decimal: Convert the annual percentage decrease to a decimal. $\newline$ To convert a percentage to a decimal, divide by $100$ . Therefore, $1\%$ becomes $0.01$ .
Determine Annual Multiplier: Determine the annual multiplier for the depreciation. $\newline$ Since the value decreases by $1\%$ , the value retains $100\% - 1\% = 99\%$ of its value each year. In decimal form, this is $0.99$ .
Use Exponential Decay Formula: Use the formula for exponential decay to find the future value of the land. $\newline$ The formula for exponential decay is $P = P_0 \times (1 - r)^t$ , where $P$ is the future value, $P_0$ is the initial value, $r$ is the rate of decrease, and $t$ is the time in years.
Substitute Values: Substitute the known values into the formula. $\newline$ $P = 65000 \times (0.99)^{20}$
Calculate Future Value: Calculate the future value of the land. $\newline$ $P = 65000 \times (0.99)^{20}$ $\newline$ Using a calculator, we find: $\newline$ $P \approx 65000 \times 0.817906937597$ $\newline$ $P \approx 53164.15094431$
Round to Nearest Dollar: Round the answer to the nearest dollar. $\newline$ The value of the land after $20$ years, rounded to the nearest dollar, is approximately $\$53,164$ .

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