A rectangular page is to contain 144 square inches of print.The margins on each side are 1 inch.Find the dimensions of the page such that the least amount of paper is used.

Question

A rectangular page is to contain 144 square inches of print.The margins on each side are 1 inch.Find the dimensions of the page such that the least amount of paper is used. ​

Quizplus · Accepted Answer

The problem can be solved using optimization techniques from calculus. Let the dimensions of the page be $x$ (width) and $y$ (height). The area of print is $144$ square inches, and since there's a $1$-inch margin on each side, the printable area dimensions become $x-2$ and $y-2$. Thus, the equation for the area of print is $(x-2)(y-2) = 144$.To minimize the paper used, we need to minimize the area of the page, which is $xy$. From the print area equation, we can express one variable in terms of the other, for example, $y = \frac{144}{x-2} + 2$.Substituting $y$ in the area function $A = xy$ gives us an equation in terms of $x$ only, which can then be differentiated to find the minimum area. However, without calculus, we can reason that since the area of print is fixed, the total area of the page (and thus the amount of paper used) will be minimized when the page is as close to a square as possible, because a square has the smallest perimeter and area for a given amount of interior space.Given the options and knowing that the print area is $144$ square inches with $1$-inch margins all around, the dimensions of the print area would be $12 \times 12$ inches (since $12 \times 12 = 144$). Adding the margins back, the total page dimensions would be $14 \times 14$ inches, making option E the correct choice.

A Rectangular Page Is to Contain 144 Square Inches of Print.The