Johnny is designing a rectangular poster to contain 18 in^2 of printing with a 3-in margin at the top and bottom and a 6-in margin at each side. what overall dimensions will minimize the amount of paper used?

The dimensions will minimize the amount of paper used is with length 9 inch and width 18 inch.

Step-by-step explanation:

Given : Johnny is designing a rectangular poster to contain 18 in² of printing with a 3-in margin at the top and bottom and a 6-in margin at each side.

To find : What overall dimensions will minimize the amount of paper used?

Solution :

Let l be the length and w be the width.

Area is

Johnny is designing a rectangular poster to contain 18 in² of printing

i.e. Area of the inner printed section   ....(1)

A 3-in margin at the top and bottom and a 6-in margin at each side.

i.e. l'=l+6 and w'=w+12

Total Area of the Poster  is ....(2)

Substitute the value of l from (1) in (2),

Differentiate w.r.t w,

For critical value put A'=0,

Differentiate again w.r.t w,

It is minimum at w=6.

Substitute in (1),

The dimensions required are

l'=l+6= 3+6=9 inch

w'=w+12= 6+12=18 inch

The dimensions will minimize the amount of paper used is with length 9 inch and width 18 inch.