Question

A certain forest covers an area of 4900 km^2
• Suppose that each year this area decreases by 8.25%. What will
the area be after 13 years?
Use the calculator provided and round your answer to the nearest square kilometer.

Answer 1

To calculate the area of the forest after 13 years with an annual decrease of 8.25%, we can use the formula for exponential decay:

\[
A_t = A_0 \times (1 - r)^t
\]

Where:
- \( A_t \) is the amount after time \( t \)
- \( A_0 \) is the initial amount (4900 km²)
- \( r \) is the rate of decrease (8.25% or 0.0825)
- \( t \) is the number of years (13)

Substituting the values into the formula:

\[
A_t = 4900 \times (1 - 0.0825)^{13}
\]

Calculating \( (1 - 0.0825) \):

\[
1 - 0.0825 = 0.9175
\]

Now, raise \( 0.9175 \) to the power of 13:

\[
(0.9175)^{13} \approx 0.327014
\]

Now, we can calculate \( A_t \):

\[
A_t \approx 4900 \times 0.327014 \approx 1609.569
\]

Finally, rounding to the nearest square kilometer:

\[
A_t \approx 1610 \text{ km}^2
\]

So, the area of the forest after 13 years will be approximately **1610 km²**.