rectangle question

Question

A rectangular page is to contain 42 square inches of printable area. The margins at the top and bottom of the page are each 1 inch, one side margin is 1 inch, and the other side margin is . What should the dimensions of the page be so that the least amount of paper is used?

Answer 1

A square contains the maximum enclosed area of any rectangle, if the margins are all the same.

dimensions are sqr42 by sqr42, about 6.48 x 6.48 inches. But the enclosed printed area would be those dimensions minus 2 or sqr42-2 by sqr42-2, about 4.48 by 4.48 inches

but you didn't specify the other side's margin. Let's call it a, and the dimensions as xy=42, with the enclosed area as (x-a)(y-2) = Area = xy-ay-2x+2a. substitute y=42/x to get Area = 42-42a/x -2x+2a. Take the derivative and set it = 0, solve for x. x=sqr14a, y=3sqr14/a or about x=3.54sqra and y=10.62/sqra. Those are the dimensions of the entire 42 square inch paper, with 3 one inch margins and the 4th margin = and unknown "a" inches.

the enclosed printed area without the blank margins would be x-1-a = 3.54sqra-1-a and y-2=10.62/sqra - 2