Answer :
To determine the daily growth rate of weeds, we need to start with the given weekly growth rate function:
[tex]\[ f(x) = 197(1.25)^x \][/tex]
This function shows that the weeds grow by 25% per week. We want to find the equivalent daily growth function and express this rate as a percentage.
Step 1: Relate the Weekly and Daily Growth Rates
The weekly growth rate is given as a factor of 1.25. We use the fact that a week has 7 days. So, we need to find a daily growth rate [tex]\( r \)[/tex] such that:
[tex]\[ (1 + r)^7 = 1.25 \][/tex]
Step 2: Solve for the Daily Growth Rate
We can solve for [tex]\( r \)[/tex] by taking the 7th root of 1.25:
[tex]\[ 1 + r = \sqrt[7]{1.25} \][/tex]
The value of [tex]\( \sqrt[7]{1.25} \)[/tex] gives us the daily growth factor. After determining this factor, subtract 1 to get the daily growth rate [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt[7]{1.25} - 1 \][/tex]
Step 3: Convert the Daily Growth Rate into a Percentage
To express the daily growth rate as a percentage, we multiply it by 100:
[tex]\[ \text{Daily Growth Rate (\%)} = (r) \times 100 \][/tex]
Final Step: Interpret the Result
Carrying out the steps above, we find that the daily growth rate as a percentage is approximately 3.24%. Therefore, the daily growth function can be written as:
[tex]\[ f(x) = 197(1.032391184710001797)^x \][/tex]
This growth rate matches approximately with one of the choices given in the question, indicating it grows at a rate of approximately [tex]\( 3.24\% \)[/tex] daily.
[tex]\[ f(x) = 197(1.25)^x \][/tex]
This function shows that the weeds grow by 25% per week. We want to find the equivalent daily growth function and express this rate as a percentage.
Step 1: Relate the Weekly and Daily Growth Rates
The weekly growth rate is given as a factor of 1.25. We use the fact that a week has 7 days. So, we need to find a daily growth rate [tex]\( r \)[/tex] such that:
[tex]\[ (1 + r)^7 = 1.25 \][/tex]
Step 2: Solve for the Daily Growth Rate
We can solve for [tex]\( r \)[/tex] by taking the 7th root of 1.25:
[tex]\[ 1 + r = \sqrt[7]{1.25} \][/tex]
The value of [tex]\( \sqrt[7]{1.25} \)[/tex] gives us the daily growth factor. After determining this factor, subtract 1 to get the daily growth rate [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt[7]{1.25} - 1 \][/tex]
Step 3: Convert the Daily Growth Rate into a Percentage
To express the daily growth rate as a percentage, we multiply it by 100:
[tex]\[ \text{Daily Growth Rate (\%)} = (r) \times 100 \][/tex]
Final Step: Interpret the Result
Carrying out the steps above, we find that the daily growth rate as a percentage is approximately 3.24%. Therefore, the daily growth function can be written as:
[tex]\[ f(x) = 197(1.032391184710001797)^x \][/tex]
This growth rate matches approximately with one of the choices given in the question, indicating it grows at a rate of approximately [tex]\( 3.24\% \)[/tex] daily.
