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A certain forest covers an area of $4400\,\text{km}^2$ . Suppose that each year this area decreases by $4.75\%$ . What will the area be after $10$ years? Use the calculator provided and round your answer to the nearest square kilometer.

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Exponential growth and decay: word problems

Full solution

Q. A certain forest covers an area of

4400\,\text{km}^2

. Suppose that each year this area decreases by

4.75\%

. What will the area be after

10

years? Use the calculator provided and round your answer to the nearest square kilometer.

Identify Change Type: Determine the type of change in the forest area. The forest area decreases by a fixed percentage each year. This indicates exponential decay.
Find Initial Values: Identify the initial value $a$ and the decay rate $r$ . $a = 4400 \, \text{km}^2$ The decay rate $r$ is $4.75\%$ , which can be written as a decimal by dividing by $100$ . $r = \frac{4.75}{100} = 0.0475$
Calculate Decay Factor: Calculate the decay factor $b$ . The decay factor $b$ is found by subtracting the decay rate from $1$ . $b = 1 - r$ $b = 1 - 0.0475$ $b = 0.9525$
Determine Years: Determine the number of years $t$ . $t = 10$ years
Use Exponential Decay Formula: Use the exponential decay formula to find the area after $t$ years. $\newline$ Exponential Decay: $A(t) = a(b)^{(t)}$ $\newline$ Substitute $4400$ for $a$ , $0.9525$ for $b$ , and $10$ for $t$ . $\newline$ $A(t) = 4400(0.9525)^{(10)}$
Calculate Area After $10$ Years: Calculate the area after $10$ years using the provided values. $\newline$ $A(t) = 4400(0.9525)^{10}$ $\newline$ Use a calculator to find $(0.9525)^{10}$ and then multiply by $4400$ . $\newline$ $(0.9525)^{10} \approx 0.6447$ $\newline$ $A(t) \approx 4400 \times 0.6447$ $\newline$ $A(t) \approx 2836.68$
Round to Nearest Square Kilometer: Round the answer to the nearest square kilometer. $A(t) \approx 2837 \, \text{km}^2$

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