Final answer:
To minimize the amount of paper used, the dimensions of the rectangular page should be a length of 9 inches and a width of 12 inches.
Step-by-step explanation:
To minimize the amount of paper used, we need to find the dimensions of the rectangular page that contain 54 square inches of print while also accounting for the margins. Let's denote the length of the page as L and the width as W. With a 3-inch margin on top and bottom, the usable height of the page is L - 2(3) = L - 6 inches. Similarly, with 2-inch margins on each side, the usable width of the page is W - 2(2) = W - 4 inches. The equation for the area of print is then (L - 6)(W - 4) = 54.
To minimize the amount of paper used, we can rewrite the equation in terms of one variable. Let's solve for L in terms of W: L = (54/(W - 4)) + 6. Now, we can substitute this expression for L into the equation for the area of print to get a quadratic equation in terms of W: (54/(W - 4))((W - 4) - 6) = 54. Simplifying this equation, we have (W - 4)(W - 10) = 54. Expanding and rearranging, we get W^2 - 14W + 44 = 0.
We can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, factoring is the most efficient method. The factors of 44 that add up to -14 are -2 and -12. Therefore, the equation factors as (W - 2)(W - 12) = 0. The possible values for W are 2 and 12. Since W represents the width of the page, we can eliminate the extraneous solution of W = 2 because it would result in a negative usable width.
Therefore, the dimensions of the page that minimize the amount of paper used are a length of (54/(12 - 4)) + 6 = 9 inches and a width of 12 inches.