Question

An on-demand printing company has monthly overhead costs of $1,900 in rent, $450 in electricity, $80 for phone service, and $230 for advertising and marketing. The printing cost is $40 per thousand pages for paper and ink. The average cost for printing x thousand pages can be represented by the function C(x)= 2660+40x/x

For a given month, if the printing company could print unlimited number of pages, what value would the average cost per thousand pages approach? What does this mean in the context of the problem?

Answer 1

$40

It means that when an infinite number of pages are printed in a month then the fixed cost will be negligible.

Explanation:

Given that:

Monthly overhead costs in rent = $1900

Costs in electricity = $450

Costs for phone service = $80

Costs for advertising and marketing = $230

Printing cost per thousand pages = $40

If number of pages printed are
`x` thousand then cost of printing for
`x` thousand pages = $40
`x`

Average cost to print
`x` thousand pages can be given as:

(Total fixed cost + Total cost of printing
`x` thousand pages) divided by
`x`

`(2660+40x)/(x)`

Now, it is given that an infinite number of pages are printed in a month, and we have to find the value that average cost would approach to.

`\lim_(x \to \infty) ((2660+40x)/(x))`

Let us solve the above limits.

`\Rightarrow \lim_(x \to \infty) (\frac{2660}x+(40x)/(x))\\\Rightarrow \lim_(x \to \infty) (\frac{2660}x)+40\\\Rightarrow 40+\lim_(x \to \infty) (\frac{2660}x)`

When

`x\rightarrow \infty\Rightarrow (2660)/(x) \rightarrow 0`

Putting the value in the limits, we get:

Average cost per thousand pages month would approach to
`\rightarrow \$40`

It means that when an infinite number of pages are printed in a month then the fixed cost will be negligible.