Sure, let's solve this step-by-step.
### Problem:
A forest covers an initial area of 2100 km². The area decreases by 3.5% each year. What will the area be after 6 years? Round your answer to the nearest square kilometer.
### Step-by-Step Solution:
1. Initial Area:
- The initial area of the forest is \(\( 2100 \)\) km².
2. Annual Decrease Rate:
- The forest area decreases by \(\( 3.5\% \)\) each year.
- To represent this as a decimal, \(\( 3.5\% = 0.035 \)\).
3. Formula for Decrease Over Time:
- The formula to calculate the area after \(\( n \)\) years, when it decreases by a certain percentage each year, is:
\(\[
A_n = A_0 \times (1 - r)^n
\]\)
Where:
- \(\( A_n \)\) is the area after \(\( n \)\) years.
- \(\( A_0 \)\) is the initial area.
- \(\( r \)\) is the annual decrease rate.
- \(\( n \)\) is the number of years.
4. Substitute the Values:
- \(\( A_0 = 2100 \)\) km²
- \(\( r = 0.035 \)\)
- \(\( n = 6 \)\)
\(\[
A_6 = 2100 \times (1 - 0.035)^6
\]\)
5. Calculate the Result:
- First calculate \(\( (1 - 0.035) \)\):
\(\[
1 - 0.035 = 0.965
\]\)
- Then raise this value to the power of 6:
\(\[
0.965^6 \approx 0.817073
\]\)
- Now multiply by the initial area:
\(\[
A_6 = 2100 \times 0.817073 \approx 1715.85
\]\)
6. Round to the Nearest Square Kilometer:
- \(\( 1715.85 \)\) rounds to \(\( 1716 \)\) km².
### Answer:
After 6 years, the area of the forest will be approximately \(\( 1716 \)\) km².
