Researchers are monitoring two bacterial cultures in identical flasks
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
Researchers are monitoring two bacterial cultures in identical flasks.
• Culture A has population A(t) (in thousands of cells) t hours after inoculation.
• Culture B has population B(t) (in thousands of cells) t hours after inoculation.
At t=4 hours the following measurements are recorded:
A(4)=12.8, A′(4)=3.2, B(4)=15.1, B′(4)=2.4.
Assume both A and B are differentiable for 0≤t≤10.
(a) Interpret the meaning of A′(4)=3.2 in the context of the bacterial culture, including appropriate units.
(b) Determine which culture is growing faster at t=4 hours. Justify your answer by explicitly comparing the relevant rates with units.
(c) Suppose the laboratory can safely handle a combined growth rate of at most 5 thousand cells per hour at t=4. Verify whether the combined growth rate of the two cultures at t=4 stays within this safety limit, and explain what your calculation implies about laboratory safety at that instant.