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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

A bacteria culture grows with constant relative growth rate. The bacteria count was 400 after 2 hours and 25,600 after 6 hours.

(a) What is the relative growth rate? Express your answer as a percentage.

(b) What was the initial size of the culture?

(c) Find an expression for the number of bacteria after t hours.

(d) Find the number of cells after 4.5 hours.

(e) Find the rate of growth after 4.5 hours.

(f) When will the population reach 50,000?

(a)

To determine

To find: The relative growth rate and to express the answer as a percentage.

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.

Given that,

The bacteria count was 400 after 2 hours. That is, P(2)=400.

The bacteria count was 25,000 after 6 hours. That is, P(6)=25,600.

The bacteria culture grows with constant relative growth rate. That is, dPdt=kP.

By using theorem as sated above,

P(t)=p0ekt (1)

Substitute P(t)=400 and t=2 in equation (1),

400=p0e2k

p0=400e2k (2)

Substitute P(t)=25,600 and t=6 in equation (1),

25600=p0e6k

p0=25600e6k (3)

Since the left hand side of equations (2) and (3) are same, the right hand side of equations (2) and (3) must be same

(b)

To determine

To find: The initial size of the culture.

(c)

To determine

To find: The number of cells after t hours is P(t).

(d)

To determine

To find: The number of cells after 4.5 hours.

(e)

To determine

To find: The rate of growth after 4.5 hours.

(f)

To determine

To find: The time at which population reach 50000.

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