Chapter 1.4, Problem 102E

Average Cost A clothing manufacturer finds that the cost of producing x shirts is 500 + 6x + 0.01x² dollars.

1. (a) Explain why the average cost per shirt is given by the rational expression

$A=\frac{500+6x+0.01x^2}{x}$

1. (b) Complete the table by calculating the average cost per shirt for the given values of x.

x	Average cost
10
20
50
100
200
500
1000

To determine

To explain: The average cost per shirt is given by rational expression $A=\frac{500+6x+0.01}{x}$.

Explanation of Solution

The cost of producing x shirts is $500+6x+0.01x^2$.

The result obtained by diving the sum of all given quantities by the total number of quantities is known as average.

$\text{Avearge}=\frac{\text{Sum}\text{of}\text{all}\text{quantities}}{\text{Number}\text{of}\text{quantities}}$

The average cost of one shirt is,

$\text{Average}\text{cost}=\frac{500+6x+0.01x^2}{x}$

Thus, the average cost per shirt is given by rational expression $A=\frac{500+6x+0.01}{x}$.

To determine

To find: Complete the table for average cost of shirt for different values of x.

Answer to Problem 102E

The table for average cost per shirt for different value of x is,

x	Average Cost
10	56.1
20	31.2
50	16.5
100	12
200	10.5
500	12
1000	16.5

Explanation of Solution

Given:

The different values of x are 10, 20, 50, 100, 200, 500 and 1000.

Calculation:

From part (a), the average cost per shirt is,

$A=\frac{500+6x+0.01x^2}{x}$ (1)

Substitute 10 for x in equation (1).

$\begin{array}{c}A=\frac{500+6\left(10\right)+0.01{\left(10\right)}^2}{10}\\ =\frac{500+60+1}{10}\\ =\frac{561}{10}\\ =56.1\end{array}$

The average cost is 56.1 dollars for 10 shirts.

Substitute 20 for x into equation (1).

$\begin{array}{c}A=\frac{500+6\left(20\right)+0.01{\left(20\right)}^2}{20}\\ =\frac{500+120+4}{20}\\ =\frac{624}{20}\\ =31.2\end{array}$

The average cost is 31.2 dollars for 20 shirts.

Substitute 50 for x into equation (1).

$\begin{array}{c}A=\frac{500+6\left(50\right)+0.01{\left(50\right)}^2}{50}\\ =\frac{500+300+25}{50}\\ =\frac{825}{50}\\ =16.5\end{array}$

The average cost is 16.5 dollars for 50 shirts.

Substitute 100 for x into equation (1).

$\begin{array}{c}A=\frac{500+6\left(100\right)+0.01{\left(100\right)}^2}{100}\\ =\frac{500+600+100}{100}\\ =\frac{1200}{100}\\ =12\end{array}$

The average cost is 12 dollars for 100 shirts.

Substitute 200 for x into equation (1).

$\begin{array}{c}A=\frac{500+6\left(200\right)+0.01{\left(200\right)}^2}{200}\\ =\frac{500+1200+400}{200}\\ =\frac{2100}{200}\\ =10.5\end{array}$

The average cost is 10.5 dollars for 200 shirts.

Substitute 500 for x into equation (1).

$\begin{array}{c}A=\frac{500+6\left(500\right)+0.01{\left(500\right)}^2}{500}\\ =\frac{500+3000+2500}{500}\\ =\frac{6000}{500}\\ =12\end{array}$

The average cost is 12 dollars for 500 shirts.

Substitute 1000 for x into equation (1).

$\begin{array}{c}A=\frac{500+6\left(1000\right)+0.01{\left(1000\right)}^2}{1000}\\ =\frac{500+6000+10000}{1000}\\ =\frac{16500}{1000}\\ =16.5\end{array}$

The average cost is 16.5 dollars for 1000 shirts.

Thus, the table for average cost per shirt for different value of x is,

x	Average Cost
10	56.1
20	31.2
50	16.5
100	12
200	10.5
500	12
1000	16.5