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 30E

(a)

To determine

To find: The value of C'(100) and interpret the result if the cost function is, C(q)=84+0.16q0.0006q2+0.000003q3.

Expert Solution

Answer to Problem 30E

The value of C'(100)=$0.13, which represents the rate of change of the cost when the 100th item is produced.

Explanation of Solution

The cost function is C(q)=84+0.16q0.0006q2+0.000003q3.

Obtain the derivative of the cost function.

C'(q)=ddx(84+0.16q0.0006q2+0.000003q3)=0.160.0012q+0.000009q2

Substitute q = 100 in C'(q) and obtain the value of C'(100).

C'(100)=0.160.0012(100)+0.000009(100)2=0.160.12+0.09=0.13

The value of C'(100)=$0.13, which represents the rate of change of the cost when the 100th item is produced.

(b)

To determine

To find: Compare the values of C'(100) and the cost of manufacturing the 101st item.

Expert Solution

Answer to Problem 30E

The cost of manufacturing the 101st pair of jeans is approximately $0.13.

Explanation of Solution

The cost function is C(q)=84+0.16q0.0006q2+0.000003q3.

From part (a), C'(100)=0.13.

The cost of manufacturing the 101st item is calculated by, C(101)C(100).

Substitute x = 100 in C(q) and find the value of C(100).

C(100)=84+0.16(100)0.0006(100)2+0.000003(100)3=84+166+3=97

Substitute x = 101 in C(q) and find the value of C(101).

C(101)=84+0.16(101)0.0006(101)2+0.000003(101)3=84+16.166.1206+3.090903=97.130303

Substitute the respective values in C(101)C(100),

C(101)C(100)=97.13030397=0.130303

The cost of manufacturing the 101st pair of jeans is approximately $0.13, which is very close to the cost of C'(100).

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