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Answer :
To solve this problem, we need to determine the function that describes the growth of the water hyacinth over time. Here's a step-by-step explanation of how we arrive at the correct function:
1. Identify Daily Growth Rate:
- The water hyacinth grows at a rate of 7% per day. This can be written as a growth factor of [tex]\(1 + 0.07 = 1.07\)[/tex].
2. Convert Daily Growth to Weekly Growth:
- Since we are interested in the growth over weeks, and there are 7 days in a week, we need to find the equivalent weekly growth rate.
- The weekly growth rate is the daily growth rate raised to the power of 7:
[tex]\[
(1.07)^7
\][/tex]
- This exponentiation gives us the compounded growth factor over a week.
3. Write the Function:
- The initial amount of water hyacinth is 150 grams.
- The amount of water hyacinth after [tex]\( t \)[/tex] weeks can be represented by multiplying the initial amount by the weekly growth factor raised to the power of [tex]\( t \)[/tex]:
[tex]\[
\text{Amount of water hyacinth} = 150 \times (1.07^7)^t
\][/tex]
Therefore, the correct function that describes the amount of water hyacinth in the testing pool [tex]\( t \)[/tex] weeks after the sample is introduced is represented by option (D):
[tex]\[
k(t) = 150 \times (1.07^7)^t
\][/tex]
This equation correctly models the exponential growth of the plant based on the weekly compound interest formula for growth.
1. Identify Daily Growth Rate:
- The water hyacinth grows at a rate of 7% per day. This can be written as a growth factor of [tex]\(1 + 0.07 = 1.07\)[/tex].
2. Convert Daily Growth to Weekly Growth:
- Since we are interested in the growth over weeks, and there are 7 days in a week, we need to find the equivalent weekly growth rate.
- The weekly growth rate is the daily growth rate raised to the power of 7:
[tex]\[
(1.07)^7
\][/tex]
- This exponentiation gives us the compounded growth factor over a week.
3. Write the Function:
- The initial amount of water hyacinth is 150 grams.
- The amount of water hyacinth after [tex]\( t \)[/tex] weeks can be represented by multiplying the initial amount by the weekly growth factor raised to the power of [tex]\( t \)[/tex]:
[tex]\[
\text{Amount of water hyacinth} = 150 \times (1.07^7)^t
\][/tex]
Therefore, the correct function that describes the amount of water hyacinth in the testing pool [tex]\( t \)[/tex] weeks after the sample is introduced is represented by option (D):
[tex]\[
k(t) = 150 \times (1.07^7)^t
\][/tex]
This equation correctly models the exponential growth of the plant based on the weekly compound interest formula for growth.
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