Answer :
To solve this problem, we need to find the correct function that represents the growth of the water hyacinth in the testing pool over a period of weeks. We know the plant grows at a rate of 7% per day, and we're interested in the growth over weeks.
1. Understand the Daily Growth Factor:
- The water hyacinth grows at a rate of 7% per day, which means each day the amount of hyacinth is multiplied by a factor of 1.07 (since 100% + 7% = 107%).
2. Calculate the Weekly Growth Factor:
- Since there are 7 days in a week, the weekly growth factor is determined by raising the daily growth factor (1.07) to the power of 7. This results in the weekly growth factor being approximately 1.6058.
3. Evaluate the Options:
- We start with 150 grams, so any function representing the growth should start with 150.
- The next part of the function should reflect the growth over a week. Based on calculations:
- Option A: [tex]\(150\left(1 + 0.07^{(1 / 7)}\right)^t\)[/tex] does not correctly reflect the growth rate.
- Option B: [tex]\(150\left(1.07^{(1 / 7)}\right)^t\)[/tex] incorrectly applies the 7% rate for a fraction of a week, not suitable.
- Option C: [tex]\(150\left(1 + 0.07^{(7)}\right)^t\)[/tex] does not correctly exponentiate the entire growth rate.
- Option D: [tex]\(150\left(1.07^{(7)}\right)^t\)[/tex] is correct because it captures the compound growth over 7 days.
4. Verify Option D:
- For [tex]\(t = 1\)[/tex] week, calculate [tex]\(150 \times 1.07^7\)[/tex].
- This operation gives us the new amount of hyacinth after one week, approximately 240.87 grams. This aligns with the earlier calculation showing that [tex]\(1.07^7 \approx 1.6058\)[/tex].
Therefore, the correct function that describes the hyacinth's growth over weeks is Option D: [tex]\(k(t) = 150 \times (1.07^{7})^t\)[/tex].
1. Understand the Daily Growth Factor:
- The water hyacinth grows at a rate of 7% per day, which means each day the amount of hyacinth is multiplied by a factor of 1.07 (since 100% + 7% = 107%).
2. Calculate the Weekly Growth Factor:
- Since there are 7 days in a week, the weekly growth factor is determined by raising the daily growth factor (1.07) to the power of 7. This results in the weekly growth factor being approximately 1.6058.
3. Evaluate the Options:
- We start with 150 grams, so any function representing the growth should start with 150.
- The next part of the function should reflect the growth over a week. Based on calculations:
- Option A: [tex]\(150\left(1 + 0.07^{(1 / 7)}\right)^t\)[/tex] does not correctly reflect the growth rate.
- Option B: [tex]\(150\left(1.07^{(1 / 7)}\right)^t\)[/tex] incorrectly applies the 7% rate for a fraction of a week, not suitable.
- Option C: [tex]\(150\left(1 + 0.07^{(7)}\right)^t\)[/tex] does not correctly exponentiate the entire growth rate.
- Option D: [tex]\(150\left(1.07^{(7)}\right)^t\)[/tex] is correct because it captures the compound growth over 7 days.
4. Verify Option D:
- For [tex]\(t = 1\)[/tex] week, calculate [tex]\(150 \times 1.07^7\)[/tex].
- This operation gives us the new amount of hyacinth after one week, approximately 240.87 grams. This aligns with the earlier calculation showing that [tex]\(1.07^7 \approx 1.6058\)[/tex].
Therefore, the correct function that describes the hyacinth's growth over weeks is Option D: [tex]\(k(t) = 150 \times (1.07^{7})^t\)[/tex].
