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Answer :
To solve this problem, we want to find the function that best represents the growth of the water hyacinth in the testing pool over time. Here's how we can approach it step-by-step:
1. Identify the Initial Situation:
You have a 150-gram sample of water hyacinth introduced into a testing pool, with a daily growth rate of 7%. This growth is compounded daily since it's given as a daily rate.
2. Convert the Daily Growth Rate to Weekly:
Since there are 7 days in a week, the weekly growth factor can be calculated by compounding the daily growth over a week. The daily growth factor is [tex]\(1 + 0.07 = 1.07\)[/tex]. Over 7 days, the weekly growth factor is [tex]\( (1.07)^7 \)[/tex].
3. Formulate the Weekly Growth Function:
To find the amount of water hyacinth after [tex]\( t \)[/tex] weeks, use the formula:
[tex]\[ \text{Amount after } t \text{ weeks} = 150 \times (1.07)^7^t \][/tex]
This accounts for the compound growth of the plant over multiple weeks.
4. Analyze the Given Options:
- Option (A): [tex]\( f(t)=150\left(1+0.07^{(1 / 7)}\right)^t \)[/tex]
- Option (B): [tex]\( g(t)=150\left(1.07^{(1 / 7)}\right)^t \)[/tex]
- Option (C): [tex]\( h(t)=150\left(1+0.07^{(7)}\right)^t \)[/tex]
- Option (D): [tex]\( k(t)=150\left(1.07^{(7)}\right)^t \)[/tex]
5. Determine the Correct Function:
Option (D), [tex]\( k(t)=150\left(1.07^{(7)}\right)^t \)[/tex], correctly represents the model where the daily growth rate of 7% is compounded over 7 days, giving the weekly growth rate. This matches our calculated model for how the water hyacinth should grow over time.
Thus, the correct function that gives the amount of water hyacinth in the testing pool [tex]\( t \)[/tex] weeks after the sample is introduced is:
[tex]\[ k(t)=150(1.07^7)^t \][/tex]
Option (D) is the correct answer.
1. Identify the Initial Situation:
You have a 150-gram sample of water hyacinth introduced into a testing pool, with a daily growth rate of 7%. This growth is compounded daily since it's given as a daily rate.
2. Convert the Daily Growth Rate to Weekly:
Since there are 7 days in a week, the weekly growth factor can be calculated by compounding the daily growth over a week. The daily growth factor is [tex]\(1 + 0.07 = 1.07\)[/tex]. Over 7 days, the weekly growth factor is [tex]\( (1.07)^7 \)[/tex].
3. Formulate the Weekly Growth Function:
To find the amount of water hyacinth after [tex]\( t \)[/tex] weeks, use the formula:
[tex]\[ \text{Amount after } t \text{ weeks} = 150 \times (1.07)^7^t \][/tex]
This accounts for the compound growth of the plant over multiple weeks.
4. Analyze the Given Options:
- Option (A): [tex]\( f(t)=150\left(1+0.07^{(1 / 7)}\right)^t \)[/tex]
- Option (B): [tex]\( g(t)=150\left(1.07^{(1 / 7)}\right)^t \)[/tex]
- Option (C): [tex]\( h(t)=150\left(1+0.07^{(7)}\right)^t \)[/tex]
- Option (D): [tex]\( k(t)=150\left(1.07^{(7)}\right)^t \)[/tex]
5. Determine the Correct Function:
Option (D), [tex]\( k(t)=150\left(1.07^{(7)}\right)^t \)[/tex], correctly represents the model where the daily growth rate of 7% is compounded over 7 days, giving the weekly growth rate. This matches our calculated model for how the water hyacinth should grow over time.
Thus, the correct function that gives the amount of water hyacinth in the testing pool [tex]\( t \)[/tex] weeks after the sample is introduced is:
[tex]\[ k(t)=150(1.07^7)^t \][/tex]
Option (D) is the correct answer.
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