College

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 the weekly weed growth: [tex]f(x)=197(1.25)^x[/tex].

Rewrite the function to show how quickly the weeds grow each day and calculate this rate as a percentage.

A. [tex]f(x)=197(1.25)^{7x}[/tex]; grows at a rate of approximately [tex]$2.5\%$[/tex] daily
B. [tex]f(x)=197\left(1.25^7\right)^x[/tex]; grows at a rate of approximately [tex]$4.77\%$[/tex] daily
C. [tex]f(x)=197(1.03)^x[/tex]; grows at a rate of approximately [tex]$0.3\%$[/tex] daily
D. [tex]f(x)=197(1.03)^{7x}[/tex]; grows at a rate of approximately [tex]$3\%$[/tex] daily

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].