Question

Lorene plans to make several open-topped boxes in which to carry plants. She makes the boxes from rectangular sheets of cardboard from

which she cuts out 4 in squares from each corner. The length of the original piece of cardboard is 8 in more than the width. If the volume of
the box is 420 in", determine the dimensions of the original piece of cardboard.
The dimensions of the original piece of cardboard are __ in by _ in.
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Answer 1

The dimensions of the original piece of cardboard are 15 inches by 23 inches.

Let's start by defining the variables we'll need for this problem:

w = width of the original piece of cardboard (in inches)

l = length of the original piece of cardboard (in inches)

h = height of the box (in inches)

We know that the length of the original piece of cardboard is 8 inches more than the width, so we can write:

l = w + 8

When Lorene cuts out 4-inch squares from each corner of the rectangular sheet, the resulting height of the box will be 4 inches. Therefore, the height of the box is:

h = 4

The base of the box will have dimensions (w - 8) inches by (l - 8) inches, since 4 inches are removed from each side of the width and length. The volume of the box is given as 420 cubic inches, so we can write:

V = (w - 8)(l - 8)(4)

Substituting the expression for l in terms of w, we get:

420 = (w - 8)(w + 8 - 8)(4)

420 = (w - 8)(w)(4)

105 = w(w - 8)

Solving for w using the quadratic formula, we get:

w = 15 or w = -7

Since w must be a positive value, we can discard the solution w = -7. Therefore, the width of the original piece of cardboard is:

w = 15 inches

Using the expression for l in terms of w, we can find the length of the original piece of cardboard:

l = w + 8 = 23 inches

Therefore, the dimensions of the original piece of cardboard are 15 inches by 23 inches.