A bacterial culture grows according to the logistic differential equation dy dt =k ,y…

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

A bacterial culture grows according to the logistic differential equation

\frac{dy}{dt}=k\,y\bigl(10^{7}-y\bigr),

where

y(t)

is the number of bacteria in the culture

t

hours after inoculation and

k

is a positive constant measured in

\text{h}^{-1}

. For many hours the environment is stable with

k=0.8

. Suddenly, fresh nutrients are added that instantaneously double the value of

k

while leaving the current population unchanged. Immediately before the nutrients are added, the culture contains

y=2.0\times10^{6}

bacteria. (a) Compute the instantaneous growth rate of the culture, in bacteria per hour, just before the nutrients are added. (b) Compute the instantaneous growth rate immediately after the addition of nutrients. (c) By what percent did the instantaneous growth rate change as a result of the nutrient addition? Round your answer to the nearest whole percent. (d) Suppose instead that the population size at the moment of nutrient addition were

y=8.0\times10^{6}

bacteria. Would doubling

k

still double the instantaneous growth rate? Justify your answer without performing any numerical calculations.

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