Question
A certain forest covers an area of \( 2400 \mathrm{~km}^{2} \). Suppose that each year this area decreases by \( 5.25 \% \). What will the area be after 7 years? Use the calculator provided and round your answer to the nearest square kilometer.
Ask by Munoz Fernandez. in Mexico
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
After 7 years, the forest area will be approximately 1646 km².
Solution
We start with an area of 2400 km² and assume that each year the forest decreases by 5.25%. This means that each year the forest retains 100% – 5.25% = 94.75% of its area, or a factor of 0.9475.
After 7 years the area can be found using the formula:
Area after 7 years = 2400 × (0.9475)^7
To compute (0.9475)^7, you can use a calculator or logarithms. Using a calculator gives approximately:
(0.9475)^7 ≈ 0.6859
Now multiply by the initial area:
2400 × 0.6859 ≈ 1646.16
Rounding to the nearest square kilometer, the area of the forest after 7 years will be approximately 1646 km².
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Simplify this solution Bonus Knowledge
To determine the area of the forest after 7 years, you can use the formula for exponential decay: \[ \text{Area after } t \text{ years} = \text{Initial Area} \times (1 - \text{Decay Rate})^t \] Given: - Initial Area = \( 2400 \ \text{km}^2 \) - Decay Rate = \( 5.25\% = 0.0525 \) - \( t = 7 \) So, the calculation will look like this: \[ \text{Area after 7 years} = 2400 \times (1 - 0.0525)^7 \] \[ = 2400 \times (0.9475)^7 \] \[ = 2400 \times 0.66764 \quad (\text{approximately}) \] \[ \approx 1602.4 \] Rounding to the nearest square kilometer, the area after 7 years will be approximately \( 1602 \ \text{km}^2 \).
