A rectangular page is to contain 98 square inches of print.

Question

The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page that will minimize the amount of paper used. (Let x represent the width of the page and let y represent the height.)

Answer 1

2 inch top & bottom margins

1 inch left & right side margins

x=width, y = height of the page

98 square inches total printed matter

what are the dimensions that minimize the total paper

total paper = xy, minimize xy

pinted part = (x-2)(y-4) =98

y=98/(x-2) + 4

xy -2y-4x+8 = 98

xy =90+2y+4x= 196/(x-2) + 8 +4x

take the derivative of xy and set = 0

(xy)'=-196/(x-2)^2 +4 = 0

-196 + 4(x-2)^2 = 0

(x-2)^2 = 196/4 = 49

x-2 = 7

x = 9 inches wide

y = 98/(9-2)+4 = 98/7+4

y = 14+4

y = 18 inches high

dimensions that minimize page area = 9 in by 18 in.

9x18 = 162 in^2 = Total page area

border, margins' area = 162-98= 64 in^2

18x2 + 9x4 - 8 = 36+36 -8 = 64 in^2

It helps to draw a diagram