06/05/2020 (2011: National Sprint Round, Problem 16)

Q: The surface area of a rectangular prism is 32 square inches. The volume of the prism is 12 cubic inches. The sum of all edge lengths is 28 inches. If the length, width and height of the prism are each increased by one inch, what is the volume of the resulting prism, in cubic inches?

A: 36

If l = length, w = width and h = height, then:

Surface Area: 2(lw + lh + wl) = 32 (square inches)

Volume: lwh = 12 (cubic inches)

Edge Length: 4(l + w + h) = 28 (inches) --> l + w + h = 7

We can assume that the side lengths are integers, and list out all sets of 3 numbers that multiply to 12:

{1, 1, 12}, {1, 2, 6}, {1, 3, 4}, {2, 2, 3}

The sum of the first set of numbers is 14, which ≠ 7. The sum of the second set of numbers is 9, which ≠ 7. The sum of the third set of numbers is 8, which is ≠ 7. The sum of the fourth set of numbers is 7, which = 7. We'll see if the dimensions fit the Surface Area equation: 2((2)(2) + (2)(3) + (2)(3)) = 2(4 + 6 + 6) = 2(16) = 32, which equals the given surface area.

Now that we know that the dimensions of the prism are 2x2x3, we  increase all of them by 1 (inch) to get the new prism with dimensions of 3x3x4. We find the volume of the new prism to be 3 * 3 * 4 = 36 cubic inches.