High School

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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 solve the problem of determining the daily growth rate of weeds in the park and rewrite the function accordingly, follow these steps:

1. Understand the original function:
- The function [tex]\( f(x) = 197(1.25)^x \)[/tex] represents the weekly growth of the weeds, where [tex]\( x \)[/tex] is the number of weeks.
- The growth rate per week is [tex]\( 25\% \)[/tex], which translates to a multiplier of [tex]\( 1.25 \)[/tex].

2. Convert weekly growth rate to daily growth rate:
- To find the daily growth rate, we need to convert the weekly growth multiplier to a daily basis.
- Since there are 7 days in a week, we compute the 7th root of the weekly multiplier [tex]\( 1.25 \)[/tex].

3. Calculate the daily growth multiplier:
- The daily growth multiplier is the 7th root of [tex]\( 1.25 \)[/tex].
- This daily growth multiplier is approximately [tex]\( 1.03239 \)[/tex].

4. Calculate the daily growth rate as a percentage:
- Convert the daily growth multiplier to a percentage.
- Subtract 1 from the daily growth multiplier, then multiply by 100 to get the percentage growth per day.
- So, the daily growth rate is approximately [tex]\( 3.24\% \)[/tex].

5. Rewriting the function for daily growth:
- The correct function that represents the daily growth of weeds is:
[tex]\[
f(x) = 197(1.03239)^{x}
\][/tex]
Here, [tex]\( x \)[/tex] represents the number of days.

6. Identify the correct option:
- Compare this new daily growth function to the given options:
- [tex]\( f(x)=197(1.03)^{7x} \)[/tex] grows at a rate of approximately [tex]\( 3\% \)[/tex] daily, which best aligns with our calculated daily growth of about [tex]\( 3.24\% \)[/tex].

The correct rewrite of the function for daily growth is approximately [tex]\( f(x) = 197(1.03)^{7x} \)[/tex], indicating a daily growth rate of about [tex]\( 3\% \)[/tex].

Thanks for taking the time to read 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. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada