The top and bottom margins of a poster are 2 ft and the side margins are each 4 ft.

Question

The top and bottom margins of a poster are 2 ft and the side margins are each 4 ft. If the area of printed material on the poster is fixed at 382 square ft, find the dimensions of the poster with the smallest area.

Let x be the width and y the height of the printed material. Find the area of the printed material as a function of x and y.

a(x, y) =

Find the total area of this poster.

Find y as a function of x, using the given value of the area of the printed material.

Rewrite the total area of this poster as a function of x.

Find the derivative of the total area of this poster as a function of x.

Find the x and y values that minimize the total area of the poster.

Find the dimensions of the poster with the smallest area.

Answer 1

xy=382

y=382/x

A = (y+4)(x+8)=(382/x+4)(x+8)

A' = (382/x +4) +(x+8)(-382/x^2)=0

382x +4x^2 -382x -8(382)= 0

x^2 = 764

x = sqr764 = 2sqr191 = about 27.64 feet wide

y = 382/2sqr191 = about 13.82 feet high