When a bactericide is added to a nutrient broth in which bacteria are growing, the bacterium population continues to grow for a while, but then stops growing and begins to decline. The size of the population at time t (hours) is b equals 8 Superscript 6 Baseline plus 8 Superscript 5 Baseline t minus 8 Superscript 4 Baseline t squared .b=86+85t−84t2. Find the growth rates at t equals 0 hours commat=0 hours, t equals 4 hours commat=4 hours, and t equals 8 hours. t=8 hours.

The growth rates at t = 0 is 8⁵.

The growth rates at t = 4 is 0.

The growth rates at t = 8 is -8⁵.

Step-by-step explanation:

The expression representing the size of the population at time t hours is:

Differentiate b (t) with respect to t to determine the growth rate as follows:

The growth rate is:

R (t) = 8⁴ (8 - 2t)

Compute the growth rates at t = 0 as follows:

Thus, the growth rates at t = 0 is 8⁵.

Compute the growth rates at t = 4 as follows:

Thus, the growth rates at t = 4 is 0.

Compute the growth rates at t = 8 as follows:

Thus, the growth rates at t = 8 is -8⁵.