Question

You are designing a rectangular poster to contain 48 in^2 of printing with a 3​-in margin at the top and bottom and a 1​-in margin at each side. What overall dimensions will minimize the amount of paper​ used?

Answer 1

I will give this a try, and if anything, maybe my strategy will suggest this problem could use some clarification. I think the ambiguity of the problem is a challenge. Anyway, 48 sq in of printing space can be achieved via 1x48, 2x24, 3x16, 4x12, or 6x8 (using strictly whole number factors). The 3+3=6 inches of added height and 1+1=2 inches of added width can be added in turn to each of these possible length and width pairs.

There are technically ten configurations of poster measurements since the problem didn’t state if it was landscape or portrait in orientation. For example, l x w, using 6x8, changes to 8x14=112 sq in. since you added 2 in of length and 6 in of height; 8x6 changes to 10x12=120 sq in. The 4x12 and 12x4 variants yield 6x18 (108 sq in) and 14x10 (140 sq in). The 3x16 and 16x3 variants yield 5x22 (110) and 18x9 (162). Most likely the 2x24 and 1x48 print areas are not optimal, but yield 4x30=120, 26x8=208, 3x54=162, and 50x7=350, and make me wonder—what exactly is getting printed?! Context matters! At the end of all this, the 1x48 uses only 162 sq in of paper while maintaining the 48 sq in of printing space. That’s my answer. Go with the 1x48–>3x54=162 sq in. Good luck!

Answer 2

To minimize the amount of paper used to create a rectangular poster with a given amount of printing and specific margins, you can use the formula for the area of a rectangle to determine the dimensions that will minimize the area of the poster.

In this case, the area of the poster must be at least 48 square inches to contain the printing, and the margins at the top and bottom must be 3 inches each, for a total of 6 inches. The margins at the sides must be 1 inch each, for a total of 2 inches. Therefore, the minimum area of the poster is 48 + 6 + 2 = 56 square inches.

To minimize the amount of paper used, we want to find the dimensions of the poster that will have the smallest possible area while still satisfying the constraints on the margins and the amount of printing. We can do this by using the formula for the area of a rectangle:

A = lw

where A is the area of the rectangle, l is the length, and w is the width.

In this case, we want to find the length and width that will minimize the area of the rectangle while still satisfying the constraints on the margins and the amount of printing. We can do this by setting the area of the rectangle equal to the minimum area we calculated above (56 square inches) and then solving for the length and width:

A = lw = 56

lw = 56

We can then use the constraints