A concrete product is being designed. It is a cube 2 ft on a side. To reduce the weight of the final product, it will be "voided" using a cardboard box. 2 in. of concrete must be maintained and there is no concrete on the bottom -- see the illustration below. What size sheet of cardboard can this shape be cut from?
A) 77 in. $ ×$ 64 in.
B) 77 in. $ ×$ 63 in.
C) 81 in. $ ×$ 64 in.
D) 81 in. $ ×$ 81 in.
E) 81 in. $ ×$ 63 in.

Question

A concrete product is being designed. It is a cube 2 ft on a side. To reduce the weight of the final product, it will be "voided" using a cardboard box. 2 in. of concrete must be maintained and there is no concrete on the bottom -- see the illustration below.      What size sheet of cardboard can this shape be cut from?
A) 77 in. $ ×$ 64 in. 
B) 77 in. $ ×$ 63 in. 
C) 81 in. $ ×$ 64 in. 
D) 81 in. $ ×$ 81 in. 
E) 81 in. $ ×$ 63 in.

Quizplus · Accepted Answer

To find the size of the cardboard sheet needed, we must first find the dimensions of the largest opening at the top of the voided cube. Since each side of the cube is 2 ft or 24 in, the length of each side of the square opening will be 20 in (24 in - 2 in on each side).

Next, we must add up the areas of the four triangles on the sides of the cube. Each of these triangles has a base of 2 in (since the void extends 2 in from each side of the cube) and a height of 22 in (since the triangle extends from the top of the cube, which is 2 in from the edge, to the center of the square opening, which is 24 in from the bottom of the cube).

The total area of the four triangles is (2 in x 22 in) x 4 = 176 in².

Therefore, the total area of the cardboard sheet needed is the area of the square opening plus the area of the four triangles:

(20 in x 20 in) + 176 in² = 576 in² + 176 in² = 752 in²

The only answer choice with an area greater than 752 in² is answer choice C, which has an area of 81 in x 64 in = 5,184 in². Therefore, the answer is C.

A Concrete Product Is Being Designed × ×× 64 In B) 77 In Being

A Concrete Product Is Being Designed $×$ 64 In
B) 77 In Being