Question

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 5 Baseline plus 8 Superscript 4 Baseline t minus 8 cubed t squared .b=85+84t−83t2. Find the growth rates at t equals 0 hours commat=0 hours, t equals 4 hours commat=4 hours, and t equals 8 hours.

Answer 1

The growth rates at t=0, t=4 and t=8 hours are 4096, 0 and -4096 respectively.

Explanation:

Consider the provided function.

`b(t)=8^5+8^4t-8^3t^2`

Now we need to find the growth rates at t=0, t=4 and t=8 hours.

Differentiate the above function.

`b'(t)=8^4-(2)8^3t`

First substitute t=0 in above function.

`b'(t)=8^4-(2)8^3(0)=4096`

Substitute t=4 in above function.

`b'(t)=8^4-(2)8^3(4)=0`

Substitute t=8 in above function.

`b'(t)=8^4-(2)8^3(8)=-4096`

Hence, the growth rates at t=0, t=4 and t=8 hours are 4096, 0 and -4096 respectively.