Question

a certain forest covers an area of 2400 km². suppose that each year this area decreases by 5.5%. 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, \( P = 2400 \), \( r = 0.055 \) (since 5.5% = 0.055), and \( t = 7 \).

Step2: Substitute the values into the formula

Substitute \( P = 2400 \), \( r = 0.055 \), and \( t = 7 \) into the formula:
\( A = 2400(1 - 0.055)^7 \)
First, calculate \( 1 - 0.055 = 0.945 \).
Then, calculate \( 0.945^7 \). Using a calculator, \( 0.945^7 \approx 0.6741 \) (rounded to four decimal places).
Now, multiply by the initial amount: \( A = 2400 \times 0.6741 \approx 1617.84 \).

Step3: Round to the nearest square kilometer

Rounding 1617.84 to the nearest whole number gives 1618.

Answer:

1618