Answer :
To solve the problem of determining the function that gives the amount of water hyacinth in the testing pool [tex]\( t \)[/tex] weeks after the sample is introduced, we need to account for the growth of the plant over time.
1. Understand the daily growth:
- The water hyacinth grows at a rate of 7% per day, which can be represented as a daily growth factor of 1.07 (since 100% + 7% = 107%, or 1.07 in decimal form).
2. Calculate the weekly growth factor:
- Since there are 7 days in a week, we need to find out how much the plant grows in one week.
- The growth factor for a single day is 1.07, so for 7 days, the growth factor is [tex]\( (1.07)^7 \)[/tex].
3. Find the weekly growth factor:
- By calculating [tex]\( (1.07)^7 \)[/tex], we determine the total growth over one week. This gives us a weekly growth factor of approximately 1.6057814764784306.
4. Express the function in terms of weeks:
- The initial sample weighs 150 grams. To find the weight after [tex]\( t \)[/tex] weeks, we multiply the initial weight by the weekly growth factor raised to the power of [tex]\( t \)[/tex]. This gives us the function:
[tex]\[
f(t) = 150 \times (1.6057814764784306)^t
\][/tex]
Comparing this function with the options given in the problem, we can see that option (D) matches:
- [tex]\( k(t) = 150(1.07^7)^t \)[/tex] simplifies to [tex]\( 150 \times (1.07^7)^t = 150 \times 1.6057814764784306^t \)[/tex].
So, the correct answer is:
(D) [tex]\( k(t) = 150(1.07^{(7)})^t \)[/tex]
1. Understand the daily growth:
- The water hyacinth grows at a rate of 7% per day, which can be represented as a daily growth factor of 1.07 (since 100% + 7% = 107%, or 1.07 in decimal form).
2. Calculate the weekly growth factor:
- Since there are 7 days in a week, we need to find out how much the plant grows in one week.
- The growth factor for a single day is 1.07, so for 7 days, the growth factor is [tex]\( (1.07)^7 \)[/tex].
3. Find the weekly growth factor:
- By calculating [tex]\( (1.07)^7 \)[/tex], we determine the total growth over one week. This gives us a weekly growth factor of approximately 1.6057814764784306.
4. Express the function in terms of weeks:
- The initial sample weighs 150 grams. To find the weight after [tex]\( t \)[/tex] weeks, we multiply the initial weight by the weekly growth factor raised to the power of [tex]\( t \)[/tex]. This gives us the function:
[tex]\[
f(t) = 150 \times (1.6057814764784306)^t
\][/tex]
Comparing this function with the options given in the problem, we can see that option (D) matches:
- [tex]\( k(t) = 150(1.07^7)^t \)[/tex] simplifies to [tex]\( 150 \times (1.07^7)^t = 150 \times 1.6057814764784306^t \)[/tex].
So, the correct answer is:
(D) [tex]\( k(t) = 150(1.07^{(7)})^t \)[/tex]
