BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.8, Problem 24E
To determine

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

Expert Solution

Answer to Problem 24E

The value of a and b are 140 and 6.

The yeast will reach the population of 140 for the long run.

Explanation of Solution

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.

0.7(20+20b)b(1+b)2=1214b+14b2=12(1+b2+2b)14b+14b2=12+12b2+24b

2b210b12=0b25b6=0

b=6 or b=1

The value cannot be negative.

Therefore, the value of b is 6.

Determine the value of a.

Substitute 6 for b in equation (6).

a=20+20(6)=140

Therefore, the value of a is 140.

Hence the value of a and b are 140 and 6.

Determine the yeast population f(t) in the long run.

f(t)=a1+be0.7t

For the population in long run take time t tends to in equation (1).

limta1+be0.7t=a1+be0.7=a1+be

=a1+b(0)=a1=a=140

Hence the yeast will reach the population of 140 in the long run.

The value of a and b are 140 and 6.

The yeast will reach the population of 140 for the long run.

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