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Answer :
To find the area of the forest after 14 years, we'll use the concept of exponential decay since the forest decreases by a fixed percentage each year. Here’s a step-by-step explanation:
1. Initial Information:
- The initial area of the forest is 3700 km².
- The annual decrease rate is 5.75%.
2. Convert Percentage to Decimal:
- The decrease rate of 5.75% must be converted to a decimal for calculations.
- This is done by dividing 5.75 by 100, which equals 0.0575.
3. Understanding Exponential Decay:
- The formula used for exponential decay is [tex]\( A = P(1 - r)^t \)[/tex], where:
- [tex]\( A \)[/tex] is the final amount (the area we want to find),
- [tex]\( P \)[/tex] is the initial amount (starting area of the forest),
- [tex]\( r \)[/tex] is the rate of decrease in decimal form,
- [tex]\( t \)[/tex] is the time period (in years).
4. Applying the Formula:
- Here, [tex]\( P = 3700 \)[/tex], [tex]\( r = 0.0575 \)[/tex], and [tex]\( t = 14 \)[/tex].
- Plug these values into the formula:
[tex]\[
A = 3700 \times (1 - 0.0575)^{14}
\][/tex]
5. Calculate the Remaining Area:
- First, subtract the rate from 1: [tex]\( 1 - 0.0575 = 0.9425 \)[/tex].
- Raise this result to the power of 14 (for 14 years): [tex]\( 0.9425^{14} \)[/tex].
- Multiply the initial area by this result: [tex]\( 3700 \times (0.9425^{14}) \)[/tex].
6. Result:
- The area of the forest after 14 years is approximately 1615 square kilometers when rounded to the nearest square kilometer.
This result shows how the area of the forest decreases over time due to a consistent reduction each year.
1. Initial Information:
- The initial area of the forest is 3700 km².
- The annual decrease rate is 5.75%.
2. Convert Percentage to Decimal:
- The decrease rate of 5.75% must be converted to a decimal for calculations.
- This is done by dividing 5.75 by 100, which equals 0.0575.
3. Understanding Exponential Decay:
- The formula used for exponential decay is [tex]\( A = P(1 - r)^t \)[/tex], where:
- [tex]\( A \)[/tex] is the final amount (the area we want to find),
- [tex]\( P \)[/tex] is the initial amount (starting area of the forest),
- [tex]\( r \)[/tex] is the rate of decrease in decimal form,
- [tex]\( t \)[/tex] is the time period (in years).
4. Applying the Formula:
- Here, [tex]\( P = 3700 \)[/tex], [tex]\( r = 0.0575 \)[/tex], and [tex]\( t = 14 \)[/tex].
- Plug these values into the formula:
[tex]\[
A = 3700 \times (1 - 0.0575)^{14}
\][/tex]
5. Calculate the Remaining Area:
- First, subtract the rate from 1: [tex]\( 1 - 0.0575 = 0.9425 \)[/tex].
- Raise this result to the power of 14 (for 14 years): [tex]\( 0.9425^{14} \)[/tex].
- Multiply the initial area by this result: [tex]\( 3700 \times (0.9425^{14}) \)[/tex].
6. Result:
- The area of the forest after 14 years is approximately 1615 square kilometers when rounded to the nearest square kilometer.
This result shows how the area of the forest decreases over time due to a consistent reduction each year.
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