Question

a certain forest covers an area of 3600 km². suppose that each year this area decreases by 7.5%. what will the area be after 7 years? use the calculator provided and round your answer to the nearest square kilometer.

Explanation:

Step1: Identify the formula for exponential decay

The formula for exponential decay is $A = P(1 - r)^t$, where $A$ is the final amount, $P$ is the initial amount, $r$ is the rate of decay (as a decimal), and $t$ is the time in years.
Here, $P = 3600$ km², $r = 0.075$ (since 7.5% = 0.075), and $t = 7$.

Step2: Substitute the values into the formula

Substitute $P = 3600$, $r = 0.075$, and $t = 7$ into the formula:
$A = 3600(1 - 0.075)^7$
First, calculate $(1 - 0.075) = 0.925$.
Then, calculate $0.925^7$. Using a calculator, $0.925^7 \approx 0.5923$.

Step3: Calculate the final area

Multiply 3600 by 0.5923:
$A = 3600 \times 0.5923 \approx 2132.28$

Step4: Round to the nearest square kilometer

Rounding 2132.28 to the nearest whole number gives 2132.

Answer:

2132