We appreciate your visit to Initially there were only 197 weeds at a park The weeds grew at a rate of tex 25 tex each week The following function represents. 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 determine how quickly the weeds grow each day, we need to convert the weekly growth rate to a daily rate. Here's how you can do it step-by-step:
1. Understand the Original Function:
The function given is [tex]\( f(x) = 197(1.25)^x \)[/tex], where [tex]\( x \)[/tex] is the number of weeks. This means that each week, the number of weeds grows by a factor of 1.25, or by 25%.
2. Convert Weekly Growth to Daily Growth:
Since we need to find the daily growth rate, we should convert the weekly factor into a daily factor. To do this, you take the 7th root of 1.25 (since there are 7 days in a week).
3. Calculate the Daily Growth Factor:
The daily growth factor can be found using the formula:
[tex]\[
\text{daily growth factor} = (1.25)^{\frac{1}{7}}
\][/tex]
4. Determine the Daily Growth Rate as a Percentage:
To express the daily growth factor as a percentage, subtract 1 and multiply by 100:
[tex]\[
\text{daily growth rate} = (\text{daily growth factor} - 1) \times 100
\][/tex]
5. Solution:
The calculation shows that the daily growth factor is approximately 1.0324, which means the percentage growth rate per day is approximately 3.24%.
Given the options, the correct answer would be:
- [tex]\( f(x)=197(1.03)^{7x}; \)[/tex] grows at a rate of approximately [tex]\( 3\% \)[/tex] daily.
This option aligns closely with the calculated growth rate when rounded to one decimal place.
1. Understand the Original Function:
The function given is [tex]\( f(x) = 197(1.25)^x \)[/tex], where [tex]\( x \)[/tex] is the number of weeks. This means that each week, the number of weeds grows by a factor of 1.25, or by 25%.
2. Convert Weekly Growth to Daily Growth:
Since we need to find the daily growth rate, we should convert the weekly factor into a daily factor. To do this, you take the 7th root of 1.25 (since there are 7 days in a week).
3. Calculate the Daily Growth Factor:
The daily growth factor can be found using the formula:
[tex]\[
\text{daily growth factor} = (1.25)^{\frac{1}{7}}
\][/tex]
4. Determine the Daily Growth Rate as a Percentage:
To express the daily growth factor as a percentage, subtract 1 and multiply by 100:
[tex]\[
\text{daily growth rate} = (\text{daily growth factor} - 1) \times 100
\][/tex]
5. Solution:
The calculation shows that the daily growth factor is approximately 1.0324, which means the percentage growth rate per day is approximately 3.24%.
Given the options, the correct answer would be:
- [tex]\( f(x)=197(1.03)^{7x}; \)[/tex] grows at a rate of approximately [tex]\( 3\% \)[/tex] daily.
This option aligns closely with the calculated growth rate when rounded to one decimal place.
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