#### To determine

**To find:** The value of C'(100) and interpret the result if the cost function is, C(q)=84+0.16q−0.0006q2+0.000003q3.

#### Answer

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

#### Explanation

The cost function is C(q)=84+0.16q−0.0006q2+0.000003q3.

Obtain the derivative of the cost function.

C'(q)=ddx(84+0.16q−0.0006q2+0.000003q3)=0.16−0.0012q+0.000009q2

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

C'(100)=0.16−0.0012(100)+0.000009(100)2=0.16−0.12+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.

#### To determine

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

#### Answer

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

#### Explanation

The cost function is C(q)=84+0.16q−0.0006q2+0.000003q3.

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

The cost of manufacturing the 101^{st} 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+16−6+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+16.16−6.1206+3.090903=97.130303

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

C(101)−C(100)=97.130303−97=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).