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Water hyacinth is an invasive plant species found in many lakes that typically grows at a rate of [tex]$7\%$[/tex] per day. As part of a study, a scientist introduces a 150-gram sample of water hyacinth into a testing pool.

Which of the following functions gives the amount of water hyacinth in the testing pool [tex]$t$[/tex] weeks after the sample is introduced? (Note: 1 week is 7 days.)

A. [tex]f(t) = 150\left(1 + 0.07^{(1/7)}\right)^t[/tex]
B. [tex]g(t) = 150\left(1.07^{(1/7)}\right)^t[/tex]
C. [tex]h(t) = 150\left(1 + 0.07^{(7)}\right)^t[/tex]
D. [tex]k(t) = 150\left(1.07^{(7)}\right)^t[/tex]

Answer :

To determine which function correctly represents the growth of the water hyacinth after [tex]\( t \)[/tex] weeks, we need to understand the relationship between daily growth and weekly growth.

1. Daily Growth Rate: The water hyacinth grows by 7% per day. This means that each day, the plant grows by a factor of [tex]\( 1.07 \)[/tex] (since 7% growth can be calculated as an increase by a multiplier of [tex]\( 1 + 0.07 = 1.07 \)[/tex]).

2. Weekly Growth Rate: Since there are 7 days in a week, we need to calculate the growth factor for an entire week. To do this, we take the daily growth rate factor and raise it to the power of 7 (the number of days in a week). So, the weekly growth rate factor is calculated as:
[tex]\[
(1.07)^7
\][/tex]

3. Growth Function: The initial sample of water hyacinth is 150 grams. Therefore, the function representing the amount of water hyacinth after [tex]\( t \)[/tex] weeks can be written in the form:
[tex]\[
k(t) = 150 \times (1.07)^7^t
\][/tex]
This shows the initial weight being multiplied by the weekly growth rate raised to the power of [tex]\( t \)[/tex].

4. Choose the Correct Option: Among the given options, option (D) [tex]\( k(t) = 150 \left(1.07^{(7)}\right)^t \)[/tex] correctly matches the derived function.

Therefore, the correct choice is option (D).

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