We appreciate your visit to Exponential and Logarithmic Functions Finding a final amount in a word problem on exponential growth or decay A certain forest covers an area of 3700. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To solve this problem of finding the area of a forest after 12 years of exponential decay, we'll follow these steps:
1. Identify the Initial Area:
The forest initially covers an area of 3700 km².
2. Understand the Rate of Decrease:
The forest's area decreases by 4.5% each year. We need to convert this percentage to a decimal for calculations:
[tex]\( \text{Decrease rate} = \frac{4.5}{100} = 0.045 \)[/tex].
3. Use the Exponential Decay Formula:
For exponential decay, the formula to calculate the remaining amount is:
[tex]\[ A = P \times (1 - r)^t \][/tex]
where:
- [tex]\( A \)[/tex] is the final area.
- [tex]\( P \)[/tex] is the initial area of the forest.
- [tex]\( r \)[/tex] is the rate of decrease in decimal form.
- [tex]\( t \)[/tex] is the time in years.
4. Plug in the Values:
Here, [tex]\( P = 3700 \)[/tex] km², [tex]\( r = 0.045 \)[/tex], and [tex]\( t = 12 \)[/tex].
So, the final area will be:
[tex]\[ A = 3700 \times (1 - 0.045)^{12} \][/tex]
5. Calculate the Final Area:
After performing the calculations:
[tex]\[ A \approx 2129.33 \text{ km}² \][/tex]
6. Round the Final Area:
Round the calculated area to the nearest square kilometer. So, the area of the forest after 12 years will be approximately 2129 km².
Therefore, after 12 years, the forest area will be approximately 2129 square kilometers.
1. Identify the Initial Area:
The forest initially covers an area of 3700 km².
2. Understand the Rate of Decrease:
The forest's area decreases by 4.5% each year. We need to convert this percentage to a decimal for calculations:
[tex]\( \text{Decrease rate} = \frac{4.5}{100} = 0.045 \)[/tex].
3. Use the Exponential Decay Formula:
For exponential decay, the formula to calculate the remaining amount is:
[tex]\[ A = P \times (1 - r)^t \][/tex]
where:
- [tex]\( A \)[/tex] is the final area.
- [tex]\( P \)[/tex] is the initial area of the forest.
- [tex]\( r \)[/tex] is the rate of decrease in decimal form.
- [tex]\( t \)[/tex] is the time in years.
4. Plug in the Values:
Here, [tex]\( P = 3700 \)[/tex] km², [tex]\( r = 0.045 \)[/tex], and [tex]\( t = 12 \)[/tex].
So, the final area will be:
[tex]\[ A = 3700 \times (1 - 0.045)^{12} \][/tex]
5. Calculate the Final Area:
After performing the calculations:
[tex]\[ A \approx 2129.33 \text{ km}² \][/tex]
6. Round the Final Area:
Round the calculated area to the nearest square kilometer. So, the area of the forest after 12 years will be approximately 2129 km².
Therefore, after 12 years, the forest area will be approximately 2129 square kilometers.
Thanks for taking the time to read Exponential and Logarithmic Functions Finding a final amount in a word problem on exponential growth or decay A certain forest covers an area of 3700. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
