Question

A rectangular page in a textbook (with width w and length l) has an area of 98in2 , top and bottom margins set at 1 inch, and left and right margins at 1 2 inch. The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the printable area?

Answer 1

Length = 14 in, width = 7 in

Explanation:

The area of the page
`= l * w = 98`

`lw = 98`

The length of the printable area is gotten by subtracting the top and bottom margins from the length of the page
`= l - 1 - 1 = l - 2`

The width of the printable area is gotten by subtracting the left and right margins from the width of the page
`w - (1)/(2)- (1)/(2) = w-1`

The area of the printable area is

`A = (l-2)(w-1) = lw - l - 2w + 2`

But
`lw = 98`,

`A = 98-l-2w+2 = 100-l-2w`

Also, from
`lw = 98`,

`l = (98)/(w)`

`A = 100 - (98)/(w) - 2w`

To maximize this area,

`(dA)/(dw) = 0`

`(98)/(w^2) - 2 = 0`

`(98)/(w^2) = 2`

`2w^2 = 98`

`w^2=49`

`w=7`

Hence,

`l = (98)/(7)=14`