AP Calculus AB · Unit 4 · Contextual Applications of Differentiation

A bacteria colony grows in a circular shape. Its radius $r(t)$, in centimeters, is given in the table for selected times $t$ hours. The number of bacteria increases at the rate $P'(t) = 60e^{0.25t}$ bacteria per hour for $0 \le t \le 10$, and there are $3000$ bacteria at $t = 0$.

(a) The area of the colony is $A(t) = \pi [r(t)]^2$. Using the average rate of change of $r$ over the interval $3 \le t \le 10$ to approximate $r'(6)$, find the rate at which the area of the colony is increasing at $t = 6$ hours. Indicate units of measure.
(b) Using the model $P'(t) = 60e^{0.25t}$, find the number of bacteria that are added to the culture during the time interval $0 \le t \le 6$. Show the integral that leads to your answer.
(c) Is the rate at which bacteria are added to the culture increasing or decreasing at $t = 4$ hours? Give a reason for your answer.
(d) Find the total number of bacteria in the culture at time $t = 10$ hours.