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Answer :
To solve the problem of determining which function represents the amount of water hyacinth in the testing pool after [tex]\( t \)[/tex] weeks, let's go through the growth process step by step:
1. Understand the Growth Rate:
- The water hyacinth grows at a rate of [tex]\( 7\% \)[/tex] per day. As a decimal, this growth rate is [tex]\( 0.07 \)[/tex] per day.
2. Convert Daily Growth Rate to Weekly Growth Rate:
- Since we need to calculate the growth over weeks and 1 week has 7 days, we need to find the weekly growth factor.
- The formula for the growth factor over a period with compounding is [tex]\((1 + \text{daily growth rate})^{\text{number of days in a week}}\)[/tex].
- So, the weekly growth rate is [tex]\((1.07)^7\)[/tex].
3. Determine the Initial Amount:
- The initial amount of water hyacinth introduced into the testing pool is 150 grams.
4. Formulate the Growth Function Over Weeks:
- The general formula for exponential growth is:
[tex]\[
\text{Amount at time } t = \text{Initial amount} \times (\text{growth factor})^t
\][/tex]
- Using the initial amount of 150 grams and the weekly growth factor calculated as [tex]\((1.07)^7\)[/tex], the function for the amount of water hyacinth after [tex]\( t \)[/tex] weeks is:
[tex]\[
k(t) = 150 \times \left((1.07)^7\right)^t
\][/tex]
5. Select the Correct Option:
- Comparing this function with the given options, we see that:
[tex]\[
k(t) = 150 \times (1.07^7)^t
\][/tex]
- This matches Option (D): [tex]\( k(t) = 150\left(1.07^{(7)}\right)^t \)[/tex].
Therefore, the correct function representing the amount of water hyacinth in the testing pool [tex]\( t \)[/tex] weeks after the sample is introduced is Option (D).
1. Understand the Growth Rate:
- The water hyacinth grows at a rate of [tex]\( 7\% \)[/tex] per day. As a decimal, this growth rate is [tex]\( 0.07 \)[/tex] per day.
2. Convert Daily Growth Rate to Weekly Growth Rate:
- Since we need to calculate the growth over weeks and 1 week has 7 days, we need to find the weekly growth factor.
- The formula for the growth factor over a period with compounding is [tex]\((1 + \text{daily growth rate})^{\text{number of days in a week}}\)[/tex].
- So, the weekly growth rate is [tex]\((1.07)^7\)[/tex].
3. Determine the Initial Amount:
- The initial amount of water hyacinth introduced into the testing pool is 150 grams.
4. Formulate the Growth Function Over Weeks:
- The general formula for exponential growth is:
[tex]\[
\text{Amount at time } t = \text{Initial amount} \times (\text{growth factor})^t
\][/tex]
- Using the initial amount of 150 grams and the weekly growth factor calculated as [tex]\((1.07)^7\)[/tex], the function for the amount of water hyacinth after [tex]\( t \)[/tex] weeks is:
[tex]\[
k(t) = 150 \times \left((1.07)^7\right)^t
\][/tex]
5. Select the Correct Option:
- Comparing this function with the given options, we see that:
[tex]\[
k(t) = 150 \times (1.07^7)^t
\][/tex]
- This matches Option (D): [tex]\( k(t) = 150\left(1.07^{(7)}\right)^t \)[/tex].
Therefore, the correct function representing the amount of water hyacinth in the testing pool [tex]\( t \)[/tex] weeks after the sample is introduced is Option (D).
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