Answer :
To determine how quickly the weeds grow each day as a percentage, we start by considering the initial situation.
1. Initial Information:
- Initially, there are 197 weeds.
- The weeds grow at a rate of 25% per week.
2. Weekly Growth Function:
- The weekly growth of weeds is represented by the function [tex]\( f(x) = 197(1.25)^x \)[/tex], where [tex]\( x \)[/tex] is the number of weeks.
3. Transition to Daily Growth:
- We need to express the growth rate on a daily basis. Since there are 7 days in a week, the daily growth function can be derived by taking the 7th root of the weekly growth rate:
[tex]\[
(1.25)^{1/7}
\][/tex]
4. Resulting Daily Growth:
- When calculated, the daily growth rate is approximately 1.03239. This means that each day, the number of weeds is multiplied by about 1.03239.
5. Daily Growth Rate as a Percentage:
- To convert the daily growth factor into a percentage increase, subtract 1 (to find just the growth part beyond the original amount) and then multiply by 100:
[tex]\[
(1.03239 - 1) \times 100 \approx 3.24\%
\][/tex]
- Therefore, the weeds grow at an approximate rate of 3.24% each day.
This step-by-step approach explains how the function representing weekly growth was adjusted to find the rate of daily growth and ultimately expressed as a percentage increase. The correct function that matches this daily growth rate is [tex]\( f(x) = 197(1.03239)^x \)[/tex], which indicates the weeds grow approximately at a rate of 3.24% daily.
1. Initial Information:
- Initially, there are 197 weeds.
- The weeds grow at a rate of 25% per week.
2. Weekly Growth Function:
- The weekly growth of weeds is represented by the function [tex]\( f(x) = 197(1.25)^x \)[/tex], where [tex]\( x \)[/tex] is the number of weeks.
3. Transition to Daily Growth:
- We need to express the growth rate on a daily basis. Since there are 7 days in a week, the daily growth function can be derived by taking the 7th root of the weekly growth rate:
[tex]\[
(1.25)^{1/7}
\][/tex]
4. Resulting Daily Growth:
- When calculated, the daily growth rate is approximately 1.03239. This means that each day, the number of weeds is multiplied by about 1.03239.
5. Daily Growth Rate as a Percentage:
- To convert the daily growth factor into a percentage increase, subtract 1 (to find just the growth part beyond the original amount) and then multiply by 100:
[tex]\[
(1.03239 - 1) \times 100 \approx 3.24\%
\][/tex]
- Therefore, the weeds grow at an approximate rate of 3.24% each day.
This step-by-step approach explains how the function representing weekly growth was adjusted to find the rate of daily growth and ultimately expressed as a percentage increase. The correct function that matches this daily growth rate is [tex]\( f(x) = 197(1.03239)^x \)[/tex], which indicates the weeds grow approximately at a rate of 3.24% daily.
