Answer :
Sure, let’s break this down step-by-step!
Problem Statement:
We’re looking to find a function that models how the water hyacinth grows in a testing pool over time. We know the plant grows at a 7% daily rate, and we want to determine this growth over multiple weeks.
Step-by-step Solution:
1. Identify Daily Growth Rate:
- The water hyacinth grows at a rate of 7% per day.
- In decimal, this is 0.07.
2. Convert Daily Growth Rate to Weekly Growth Rate:
- It’s important to understand how much the water hyacinth grows over a week.
- One week consists of 7 days.
- We will calculate the effective weekly growth rate by compounding the daily growth rate over 7 days.
[tex]\[
\text{Weekly growth rate} = (1 + \text{daily growth rate})^{7} - 1
\][/tex]
3. Using the Calculation:
- Plug in the daily growth rate:
[tex]\[
(1 + 0.07)^{7} - 1 \approx 0.6057814764784306
\][/tex]
- This means the water hyacinth increases its mass by approximately 60.58% every week.
4. Model the Growth Function:
- We start with a 150-gram sample.
- The function should model this growth over [tex]\( t \)[/tex] weeks.
- You use the formula:
[tex]\[
f(t) = 150 \times (1 + \text{weekly growth rate})^t
\][/tex]
- Substitute the weekly growth rate:
[tex]\[
f(t) = 150 \times (1.6057814764784306)^t
\][/tex]
5. Conclusion:
From the available options in the question, none match exactly with the correct compounded growth model given our calculated weekly growth rate. Therefore, the accurate function isn't listed among the choices provided, but we've derived the correct model based on compounding principles.
This step-by-step solution explains how we arrived at the weekly growth rate and how it applies to modeling the growth of water hyacinth over time.
Problem Statement:
We’re looking to find a function that models how the water hyacinth grows in a testing pool over time. We know the plant grows at a 7% daily rate, and we want to determine this growth over multiple weeks.
Step-by-step Solution:
1. Identify Daily Growth Rate:
- The water hyacinth grows at a rate of 7% per day.
- In decimal, this is 0.07.
2. Convert Daily Growth Rate to Weekly Growth Rate:
- It’s important to understand how much the water hyacinth grows over a week.
- One week consists of 7 days.
- We will calculate the effective weekly growth rate by compounding the daily growth rate over 7 days.
[tex]\[
\text{Weekly growth rate} = (1 + \text{daily growth rate})^{7} - 1
\][/tex]
3. Using the Calculation:
- Plug in the daily growth rate:
[tex]\[
(1 + 0.07)^{7} - 1 \approx 0.6057814764784306
\][/tex]
- This means the water hyacinth increases its mass by approximately 60.58% every week.
4. Model the Growth Function:
- We start with a 150-gram sample.
- The function should model this growth over [tex]\( t \)[/tex] weeks.
- You use the formula:
[tex]\[
f(t) = 150 \times (1 + \text{weekly growth rate})^t
\][/tex]
- Substitute the weekly growth rate:
[tex]\[
f(t) = 150 \times (1.6057814764784306)^t
\][/tex]
5. Conclusion:
From the available options in the question, none match exactly with the correct compounded growth model given our calculated weekly growth rate. Therefore, the accurate function isn't listed among the choices provided, but we've derived the correct model based on compounding principles.
This step-by-step solution explains how we arrived at the weekly growth rate and how it applies to modeling the growth of water hyacinth over time.
