Question

A rectangular page is to contain 24 square inches of print. The margins at the top and the bottom of page are to be 1.5 inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used? Justify your answer.

Answer 1

Answer:

  9 inches high by 6 inches wide

Step-by-step explanation:

Page area is minimized when the aspect ratio is the same as the ratio of margin widths. Here, that means the ratio of height to width should be ...

  1.5 : 1 = 3 : 2 = 6 : 4

The latter dimensions for the print area make it be 24 square inches. When 3" of vertical margin and 2 inches of horizontal margin are added, the page size should be 9 inches high by 6 inches wide.

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If the width of the paper is x inches, the height is ...

  h = 24/(x -2) +3

and the overall area is the product of length and width:

  paper area = x(24/(x -2) +3)

Differentiating area with respect to x, we can find the minimum when the derivative is zero.

  d(area)/dx = 24/(x -2) +3 -24x/(x -2)² = 0

  -3(x² -4x -12)/(x -2)² = 0 . . . . . combine terms

The numerator factors as ...

  (x -6)(x +2) = 0

So, the paper width giving minimum area is 6 inches. The height is ...

  h =  24(6 -2) +3 = 9 . . . . inches

The least paper is used when the page dimensions are 6 inches wide by 9 inches high.