Question

You are designing a rectangular poster to contain 50 in2 of printing with a 4​-in margin at the top and bottom and a 2​-in margin at each side.

What overall dimensions will minimize the amount of paper​ used?

Answer 1

W = 7.53 inches

H = 30.14 inches

Step-by-step explanation:

Printing area is:

`A_p=W_o*H_o=50 in^2`

Solving for
`H_o`:

`H_o=50/W_o`

The total area of the paper is:

`A_t=(W_o+4)*(H_o+16)` Replacing
`H_o`:

`A_t=(W_o+4)*(50/W_o+16)`

If we derivate and equal zero to find the minimum:

`A_t'=(50/W_o+16)+(W_o+4)*(-50/W_o^2)`

`0=(50/W_o+16)+(W_o+4)*(-50/W_o^2)`

`0=16-200/W_o^2`

`W_o=√(200/16)`

`W_o=3.53-inch` So,

`H_o=14.14-inch`

The final dimensions of the paper are:

W = 3.53 + 4=7.53-inch

H = 14.14 + 16= 30.14-inch