To calculate the forest area after 7 years with a 3% annual decrease, we use the exponential decay formula. After 7 years, the area will be approximately 2612.15 km².
The student's question deals with the application of percentage decrease over a period of time, specifically looking at how a forest area decreases each year. To determine the forest area after 7 years with an annual decrease of 3%, we use the formula for exponential decay:
\(A = P(1 - r)^t\)
Where:
- \(A\) is the amount of area left after time \(t\)
- \(P\) is the initial area (3200 km²)
- \(r\) is the rate of decrease (3% or 0.03)
- \(t\) is the time in years (7 years)
Step 1: Substitute the known values into the formula.
\(A = 3200(1 - 0.03)^7\)
Step 2: Calculate the decrease for one year (1 - 0.03 = 0.97).
\(A = 3200(0.97)^7\)
Step 3: Raise 0.97 to the power of 7.
\(A = 3200(0.97)^7 = 3200(0.81629788)\)
Step 4: Multiply the initial area by the decay factor.
\(A = 3200 \times 0.81629788 \approx 2612.1532\)
The forest area will be approximately 2612.15 km² after 7 years, considering the 3% annual decrease.