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