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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells.

(a) Find the number of bacteria after t hours.

(b) Find the number of bacteria after 4 hours.

(c) Find the rate of growth after 4 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 200 cells. That is, P(0)=200.

After half hour the population increased to 360, P(12)=360.

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=12 in the above equation,

P(12)=P(0)ek(12) (1)

Substitute P(0)=200 and P(12)=360 in equation (1),

360=200ek2360200=ek21

(b)

To determine

To find: The number of bacteria after 4 hours is P(4).

(c)

To determine

To find: The rate of growth after 4 hours.

(d)

To determine

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

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