Question

A clothing manufacturer finds that the cost of producing x shirts is $500+6x+0.01x^2$dollars.

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

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

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

Short answer

(a) We have

$\begin{array}{l}A=\frac{\text{Sumofalltheshirts}}{\text{Totalnumberofshirts}}\\ \Rightarrow A=\frac{500+6x+0.01x^2}{x}\end{array}$

(b) The completed table is given by

Step-by-step solution

Part a. Step 1. Apply the concept of average.

The averageof n values is given by the ratio of sum of all the values and n (total number of values).

Part a. Step 2. Analyze the given situation.

Given cost of producing x shirts is

$500+6x+0.01x^2$

Part a. Step 3.  Average.

Number of shirts is x and the sum of costs of all is$500+6x+0.01x^2$

So, the average cost per shirt is given by

$\begin{array}{l}A=\frac{\text{Sumofalltheshirts}}{\text{Totalnumberofshirts}}\\ \Rightarrow A=\frac{500+6x+0.01x^2}{x}\end{array}$

Hence proved.

Part b. Step 1. Apply the concept of average.

The averageof n values is given by the ratio of sum of all the values and n (total number of values).

Part b. Step 2. From step (a).

The average cost per shirt is given by the rational expression

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

Part b. Step 3. Complete the table.

$x$	$x^2$	$6x$	$0.01x^2$	$500+6x+0.01x^2$	$A=\frac{500+6x+0.01x^2}{x}$
10	100	60	1	561	56.1
20	400	120	4	624	31.2
50	2500	300	25	825	16.5
100	10000	600	100	1200	12
200	40000	1200	400	2100	10.5
500	250000	3000	2500	6000	12
1000	1000000	6000	10000	16500	16.5

So, the completed table is:

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