Question

How to solve this problem?

Answer 1

(a) `(30x)/(x-4)+2x`; (b) `x>4` and (c) `11.75` inches

Explanation:

As width of page is `x` and height of page is `y` and

top and bottom margins are `1"` and margins on each side are `2"`

the dimensions of print area are `x-4` and `y-2`

and as print area is `30in^2`, we have `y-2=30/(x-4)` and `y=30/(x-4)+2`

(a) Hence area of page `A` in terms of `x` is

`x(30/(x-4)+2)` or `(30x)/(x-4)+2x`

(b) We ought to have `x>4` so that margins of `2"` on either side are there.

(c) The graph of area vs. `x` appears as

graph{(30x)/(x-4)+2x [-5, 25, 43.32, 83.32]}

and area is minimum at around `x~=11` or `12` nches.

Using calculus given `A=(30x)/(x-4)+2x`, we have

`(dA)/(dx)=30((x-4-x)/(x-4)^2)+2=(-120+2(x-4)^2)/(x-4)^2`

and this is zero when `(x-4)^2=60` and `x=sqrt60+4~=11.75`