   Chapter 3.8, Problem 3E

Chapter
Section
Textbook Problem

A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420.(a) Find an expression for the number of bacteria after t hour.(b) Find the number of bacteria after 3 hours.(c) Find the rate of growth after 3 hours.(d) When will the population reach 10.000?

(a)

To determine

To find: The number of bacteria after t hours.

Explanation

Theorem used:

“The only solutions of the differential equation dydt=ky are the exponential functions

y(t)=y(0)ekt.”

Calculation:

Let P(t) be the number of bacteria after t hours.

The bacteria culture initially contains 100 cells. That is, P(0)=100.

After one hour the population increased to 420, P(1)=420.

Since the rate of the population of the bacteria is proportional to its size, dPdtP(t).

dPdt=kP, where k is proportional constant.

By theorem stated above, P(t)=P(0)ekt.

Substitute t=1 in the above equation,

P(1)=P(0)ek(1) (1)

Substitute P(0)=100 and P(1)=420 in equation (1),

420=100ek420100=ek4

(b)

To determine

To find: The number of bacteria after 3 hours is P(3).

(c)

To determine

To find: The rate of growth after 3 hours.

(d)

To determine

To find: The time at which population reach 10,000.

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