Answer :
To determine the correct function that represents the growth of the water hyacinth in the testing pool over time, let's break down the information and calculations:
1. Initial Information:
- You start with 150 grams of water hyacinth.
- The growth rate of water hyacinth is 7% per day.
- 1 week consists of 7 days.
2. Goal:
- Find the function that gives the amount of water hyacinth, [tex]\( t \)[/tex] weeks after it's introduced.
3. Growth Rate Calculation:
- Daily growth factor is [tex]\( 1 + 0.07 = 1.07 \)[/tex] because the plant grows by 7% each day.
- To find the growth factor for a week (7 days), we calculate [tex]\( (1.07)^7 \)[/tex].
4. Weekly Growth Factor:
- Calculating [tex]\( (1.07)^7 \)[/tex] gives approximately 1.6057814764784306. This is the weekly growth factor, meaning that at the end of each week, the amount of water hyacinth is multiplied by this factor.
5. Function Formulation:
- We use the initial mass of the water hyacinth (150 grams) and multiply it by the weekly growth factor raised to the power of [tex]\( t \)[/tex] weeks.
- The function is: [tex]\( k(t) = 150 \times (1.07^7)^t \)[/tex].
6. Matching with Given Options:
- Option (D) is [tex]\( k(t) = 150 \times (1.07^{7})^t \)[/tex].
Therefore, the correct function is option (D):
[tex]\[ k(t) = 150 \left(1.07^{7}\right)^t \][/tex]
This function correctly represents the growth of the water hyacinth in the pool over [tex]\( t \)[/tex] weeks.
1. Initial Information:
- You start with 150 grams of water hyacinth.
- The growth rate of water hyacinth is 7% per day.
- 1 week consists of 7 days.
2. Goal:
- Find the function that gives the amount of water hyacinth, [tex]\( t \)[/tex] weeks after it's introduced.
3. Growth Rate Calculation:
- Daily growth factor is [tex]\( 1 + 0.07 = 1.07 \)[/tex] because the plant grows by 7% each day.
- To find the growth factor for a week (7 days), we calculate [tex]\( (1.07)^7 \)[/tex].
4. Weekly Growth Factor:
- Calculating [tex]\( (1.07)^7 \)[/tex] gives approximately 1.6057814764784306. This is the weekly growth factor, meaning that at the end of each week, the amount of water hyacinth is multiplied by this factor.
5. Function Formulation:
- We use the initial mass of the water hyacinth (150 grams) and multiply it by the weekly growth factor raised to the power of [tex]\( t \)[/tex] weeks.
- The function is: [tex]\( k(t) = 150 \times (1.07^7)^t \)[/tex].
6. Matching with Given Options:
- Option (D) is [tex]\( k(t) = 150 \times (1.07^{7})^t \)[/tex].
Therefore, the correct function is option (D):
[tex]\[ k(t) = 150 \left(1.07^{7}\right)^t \][/tex]
This function correctly represents the growth of the water hyacinth in the pool over [tex]\( t \)[/tex] weeks.