Question

a certain forest covers an area of 4800 km². suppose that each year this area decreases by 4%. 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, \( P = 4800 \), \( r = 0.04 \) (since 4% = 0.04), and \( t = 7 \).

Step2: Substitute the values into the formula

Substitute \( P = 4800 \), \( r = 0.04 \), and \( t = 7 \) into the formula:
\( A = 4800(1 - 0.04)^7 \)
First, calculate \( 1 - 0.04 = 0.96 \).
Then, calculate \( 0.96^7 \). Using a calculator, \( 0.96^7 \approx 0.7514 \) (rounded to four decimal places).

Step3: Calculate the final amount

Multiply \( 4800 \) by \( 0.7514 \):
\( A = 4800 \times 0.7514 \approx 3606.72 \)
Round to the nearest square kilometer, so \( A \approx 3607 \).

Answer:

3607