Answer :
To determine how quickly the weeds grow each day, we need to convert the weekly growth rate into a daily growth rate. The problem states that the weekly growth rate is 25%. This means the function for weekly growth is [tex]\( f(x) = 197(1.25)^x \)[/tex].
Here are the steps to rewrite the function for daily growth:
1. Calculate the Daily Growth Rate:
To find the daily growth rate, we take the 7th root of the weekly growth factor. The weekly growth factor is [tex]\( 1.25 \)[/tex], which accounts for a 25% increase over the week.
Mathematically, calculate the daily growth factor as:
[tex]\[
\text{Daily Growth Factor} = 1.25^{1/7}
\][/tex]
2. Convert to Percentage:
The daily growth rate in percentage is calculated by subtracting 1 from the daily growth factor and then converting it to a percentage:
[tex]\[
\text{Daily Growth Percentage} = (\text{Daily Growth Factor} - 1) \times 100
\][/tex]
3. Interpret the Results:
After performing these calculations, the daily growth factor is approximately [tex]\( 1.0324 \)[/tex], which translates to a daily growth rate of about 3.24%.
Therefore, the function that represents daily weed growth is:
[tex]\( f(x) = 197(1.0324)^x \)[/tex]
The weeds grow at a rate of approximately 3.24% daily.
This matches the option:
[tex]\( f(x)=197(1.03)^{7x} \)[/tex]; grows at a rate of approximately 3% daily.
However, the daily function using a more precise value would be [tex]\( f(x) = 197(1.0324)^x \)[/tex].
Here are the steps to rewrite the function for daily growth:
1. Calculate the Daily Growth Rate:
To find the daily growth rate, we take the 7th root of the weekly growth factor. The weekly growth factor is [tex]\( 1.25 \)[/tex], which accounts for a 25% increase over the week.
Mathematically, calculate the daily growth factor as:
[tex]\[
\text{Daily Growth Factor} = 1.25^{1/7}
\][/tex]
2. Convert to Percentage:
The daily growth rate in percentage is calculated by subtracting 1 from the daily growth factor and then converting it to a percentage:
[tex]\[
\text{Daily Growth Percentage} = (\text{Daily Growth Factor} - 1) \times 100
\][/tex]
3. Interpret the Results:
After performing these calculations, the daily growth factor is approximately [tex]\( 1.0324 \)[/tex], which translates to a daily growth rate of about 3.24%.
Therefore, the function that represents daily weed growth is:
[tex]\( f(x) = 197(1.0324)^x \)[/tex]
The weeds grow at a rate of approximately 3.24% daily.
This matches the option:
[tex]\( f(x)=197(1.03)^{7x} \)[/tex]; grows at a rate of approximately 3% daily.
However, the daily function using a more precise value would be [tex]\( f(x) = 197(1.0324)^x \)[/tex].
