Question

Suppose that an open box is to be made from a square sheet of cardboard by cutting out squares from each corner as shown and then folding along the dotted lines. If the box is to have a volume of 162 cubic inches, find the original dimensions of the sheet of cardboard.

Answer 1

To determine the original dimensions of the square cardboard sheet for making a box with a volume of 162 cubic inches, one must set up a cubic equation where the size of the cutout squares is x, and the side of the original square cardboard is s. By solving this equation, the lengths can be determined and confirmed by ensuring they yield the desired volume.

Step-by-step explanation:

To solve for the original dimensions of the cardboard when constructing an open box with a volume of 162 cubic inches, we need to let the size of the square cutouts be x and the side of the original square cardboard be s. After the squares are cut out and the sides are folded up, the box's length and width will each be reduced by 2x, and the height will be x. Hence, the volume of the box can be represented by the equation:

V = (s - 2x)^2 * x = 162 cubic inches

This is a cubic equation in terms of x, and solving this equation gives us the value of x. Then, the original dimensions of the cardboard (side s) can be found by adding twice the size of the cutout squares to the length and width of the base of the box, given by (s - 2x).

For example, if s - 2x = 9 (the base length and width), and x = 2 (the height), then the original cardboard side s would be s = 9 + 2(2) = 13 inches. Ensure that the obtained value of x, when plugged back into the volume equation, gives 162 cubic inches to confirm the correctness.