   Chapter 3.7, Problem 26E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function n = f ( t ) = a 1 + b e − 0.7 t where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of a and b. According to this model, what happens to the yeast population in the long run?

To determine

To find: The value of a and b for the given model expression and the yeast population after a long run.

Explanation

Given:

The population is modelled by a function as below.

n=f(t)=a1+be0.7t (1)

Differentiate (1) with respect to t.

f'(t)=a(1+be0.7t)2(0.7be0.7t)

f'(t)=0.7abe0.7t(1+be0.7t)2 (2)

At time t=0, the population is 20 cells.

f(0)=20 (3)

Increase rate of population with respect to time is given as below.

f'(0)=12 (4)

Calculation:

Equate the equation (4) and (2).

Substitute 0 for t in equation (2).

f'(0)=0.7abe0.7(0)(1+be0.7(0))2=0.7abe0(1+be0)2

f'(0)=0.7ab(1+b)2 (5)

Equate equation (4) and (3).

a1+b=20a=20(1+b)

a=20+20b (6)

Determine the value of b.

Equate equation (4) and (5).

0.7ab(1+b)2=12

Substitute 20+20b for a from equation (6) in the above equation

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