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Answer :
Sure! Let's solve the problem step-by-step to determine which function represents the growth of the water hyacinth in the testing pool.
### Problem Understanding:
1. Initial Situation: You start with a 150-gram sample of water hyacinth.
2. Growth Rate: The plant grows at a rate of 7% per day.
3. Time Frame: We are interested in finding the amount of water hyacinth after [tex]\( t \)[/tex] weeks (1 week = 7 days).
### Objective:
We need to find a function that describes the amount of water hyacinth in the pool after [tex]\( t \)[/tex] weeks.
### Steps to Determine the Function:
1. Daily Growth Factor:
- If a quantity grows by 7% per day, the daily growth factor is [tex]\( 1 + 0.07 = 1.07 \)[/tex].
2. Weekly Time Conversion:
- Since 1 week has 7 days, after [tex]\( t \)[/tex] weeks, the number of days is [tex]\( 7t \)[/tex].
3. Growth Formula:
- The general formula for exponential growth is given by:
[tex]\[
\text{Final amount} = \text{Initial amount} \times (\text{growth factor})^{\text{number of periods}}
\][/tex]
- For our case:
[tex]\[
\text{Amount after } t \text{ weeks} = 150 \times (1.07)^{7t}
\][/tex]
4. Choosing the Correct Function:
- Based on the analysis, the correct function that models the situation is:
[tex]\[
k(t) = 150 \times (1.07)^{7t}
\][/tex]
### Matching to Options:
From the choices given:
- Option D matches our derived formula:
[tex]\[
k(t) = 150 \times (1.07)^{7t}
\][/tex]
Therefore, the function that accurately represents the growth of the water hyacinth in the testing pool over [tex]\( t \)[/tex] weeks is Option D.
### Problem Understanding:
1. Initial Situation: You start with a 150-gram sample of water hyacinth.
2. Growth Rate: The plant grows at a rate of 7% per day.
3. Time Frame: We are interested in finding the amount of water hyacinth after [tex]\( t \)[/tex] weeks (1 week = 7 days).
### Objective:
We need to find a function that describes the amount of water hyacinth in the pool after [tex]\( t \)[/tex] weeks.
### Steps to Determine the Function:
1. Daily Growth Factor:
- If a quantity grows by 7% per day, the daily growth factor is [tex]\( 1 + 0.07 = 1.07 \)[/tex].
2. Weekly Time Conversion:
- Since 1 week has 7 days, after [tex]\( t \)[/tex] weeks, the number of days is [tex]\( 7t \)[/tex].
3. Growth Formula:
- The general formula for exponential growth is given by:
[tex]\[
\text{Final amount} = \text{Initial amount} \times (\text{growth factor})^{\text{number of periods}}
\][/tex]
- For our case:
[tex]\[
\text{Amount after } t \text{ weeks} = 150 \times (1.07)^{7t}
\][/tex]
4. Choosing the Correct Function:
- Based on the analysis, the correct function that models the situation is:
[tex]\[
k(t) = 150 \times (1.07)^{7t}
\][/tex]
### Matching to Options:
From the choices given:
- Option D matches our derived formula:
[tex]\[
k(t) = 150 \times (1.07)^{7t}
\][/tex]
Therefore, the function that accurately represents the growth of the water hyacinth in the testing pool over [tex]\( t \)[/tex] weeks is Option D.
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