The top and bottom margins of a poster are 2 cm and the side margins are each 6 cm. If the area of printed material on the poster is fixed at 386 square centimeters

Question

Find the dimensions of the poster with the smallest area.

Width = 

Height = 

Please help!! Thank you!

Answer 1

It is easier to solve first for the width and height of the printable area, then use that to solve for the width and height of the poster.

The area of the printable area is wh

wh = 486

The poster has width w + 2*6 (as there are margins on both sides) and height h + 2 * 2

We seek to minimize the area of the poster =

(w+ 2 * 6)(h + 2 * 2) or (w + 12)(h + 4)

As h = 486, we may substitute h = 486/w

Then, A = (w + 12)(486/w + 4) is the size of the poster which we seek to minimize.

(w + 12)(486/w + 4) = 486 + 4w + 5832/w + 48 =

4w + 534 + 5832/w

We minimize this by calculating

dA/dw = 4 - 5832/w^2 and setting it equal to 0

4 - 5832/w^2 = 0

4w^2 = 5832

w^2 = 5832/4

w = √1458 = 27√2

h = 486/(27√2) = 9√2

(You could either calculate the square root of 1458 by prime factorization or note that 1458 is even but not divisible by 4, so you factor out 2, then either recognize that √729 = 27 or calculate and notice that it works out evenly).

Note how h and w have the same ratio as the margins.

Then, since the problem asks for the width and height of the poster,

Width = 27√2 + 12

Height = 9√2 + 4