A laboratory is cultivating a yeast population (in hundreds of cells)

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

Problem

A laboratory is cultivating a yeast population

Y(t)

(in hundreds of cells). For the first

5

hours the growth of the yeast is uninhibited, and

Y

satisfies

\displaystyle \frac{dY}{dt}=k_1Y

with

k_1>0

. At

t=5

hours, the temperature of the incubator is lowered, and from that moment on the population satisfies

\displaystyle \frac{dY}{dt}=k_2Y

with

k_2<0

. The following observations are recorded: •

Y(0)=200

. •

Y(5)=3200

. •

Y(8)=2000

. (a) Determine the numerical value of

k_1

(in

\text{hr}^{-1}

). (b) Obtain an explicit expression for

Y(t)

for

0\le t\le5

. (c) Use the data provided for

t=5

and

t=8

to find the numerical value of

k_2

(in

\text{hr}^{-1}

). (d) Write a piecewise-defined expression for

Y(t)

that is valid for all

t\ge0

. (e) Estimate, to the nearest tenth of an hour, the time after

t=5

when the yeast population will decline to 500 cells.