Four corners are cut from a rectangular piece of cardboard that measures 5 ft by 3 ft. The cuts are x feet from the corners, as shown in the figure below. After the cuts are made, the sides of the rectangle are folded to form an open box. The area of the bottom of the box is 12 ft^2.

Question

Four corners are cut from a rectangular piece of cardboard that measures 5 ft by 3 ft. The cuts are x feet from the corners, as shown in the figure below. After the cuts are made, the sides of the rectangle are folded to form an open box. The area of the bottom of the box is 12 ft^2.

What two equations represent the area, A, of the bottom of the box?

Answer 1

a = (5-2x)(3-2x)

a = 12

Answer 2

Well, if I were a box, I'd be an open box because I'm not good at keeping secrets. Anyway, let's unfold this problem together!

To find the area of the bottom of the box, we need to determine the dimensions of the rectangle when it's folded.

When you cut x feet from each corner, the length and width of the rectangle get reduced by 2x feet (since it's cut from both sides of each corner). Therefore, the length becomes (5 - 2x) ft, and the width becomes (3 - 2x) ft.

Now, to find the area of the bottom of the box (A), we multiply the length and width together:

A = (5 - 2x)(3 - 2x)

And since we're looking for the area to be 12 ft^2, we can set this equation equal to 12:

(5 - 2x)(3 - 2x) = 12

So, the two equations that represent the area of the bottom of the box are:

A = (5 - 2x)(3 - 2x)
(5 - 2x)(3 - 2x) = 12

Now, if only finding x was as fun as finding the humor in this question!

Answer 3

Let's first find the dimensions of the box after the cuts are made.

Since the length of the cardboard is 5 ft and the cuts are x feet from the corners, the length of the bottom of the box will be (5 - 2x) ft.
Similarly, since the width of the cardboard is 3 ft and the cuts are x feet from the corners, the width of the bottom of the box will be (3 - 2x) ft.

The area, A, of the bottom of the box is given by the formula A = length × width. So, the two equations representing the area of the bottom of the box are:

1. A = (5 - 2x)(3 - 2x)
2. A = 12 ft^2

The first equation represents the area of the bottom of the box in terms of x, while the second equation represents the given area of the bottom of the box, which is 12 ft^2.

Answer 4

To find the two equations representing the area, A, of the bottom of the box, let's start by visualizing the problem.

We have a rectangular piece of cardboard with dimensions 5 ft by 3 ft. Four corners are cut out of this rectangle, and each cut is x feet from the corner. After folding the sides, we form an open box.

Let's represent the length of the box as l (in feet) and the width as w (in feet). The height of the box will be x feet.

To find the dimensions of the bottom of the box, we need to subtract twice the value of x from both the length and the width.

The length of the bottom of the box will be: l - 2x
The width of the bottom of the box will be: w - 2x

Now, we know that the area of the bottom of the box is 12 ft². The formula for the area of a rectangle is A = l * w.

Substituting the values, we have:
A = (l - 2x) * (w - 2x)

This equation represents one of the possible equations for the area, A, of the bottom of the box.

Now, we can simplify this equation by expanding it:
A = lw - 2lx - 2wx + 4x²

Simplifying further:
A = lw - 2lx - 2wx + 4x²

This equation represents the second possible equation for the area, A, of the bottom of the box.

So, the two equations that represent the area of the bottom of the box are:
A = (l - 2x) * (w - 2x)
A = lw - 2lx - 2wx + 4x²