# Thread: Area of a Rectangular Box Cut Out from a Square Sheet

1. ## Area of a Rectangular Box Cut Out from a Square Sheet

In the image ( see below), the figure represents a single sheet of cardboard.

It can be folded along the dotted lines into a rectangular box with square sides, a longest side of 12 inches, and volume of 300 cubic inches.

If the piece was cut from a square sheet of cardboard, what is the minimum required area of the sheet of cardboard (in square inches)?

Note the figure is not drawn to scale.

2. Originally Posted by fcabanski
In the image ( see below), the figure represents a single sheet of cardboard.

It can be folded along the dotted lines into a rectangular box with square sides, a longest side of 12 inches, and volume of 300 cubic inches.

If the piece was cut from a square sheet of cardboard, what is the minimum required area of the sheet of cardboard (in square inches)?

Note the figure is not drawn to scale.

Hi, what have you tried?

Let the flaps at the end be squares with side length x. Then the original cardboard must have had dimensions of (12 + 2x) by (12 + 2x). What can you do with that?

3. Since the sides are square those flaps, and the width of all the segments, must be 5.

That checks with the volume = 5 * 5 * 12 = 300.

Then the width of the cut out is 12 + 5 + 5 = 22.

The length (height) is 20.

The minimum sides for that square are 22 x 22, for an area of 484.

Checking if that is correct.

4. How did you choose the width being 5inches?

5. The edges of the rectangular box have to be square. So the height of each segment, as well as the flaps, have to be 5. It's the only number that works with 12 to make a 300 cubic inch volume.

5 x 5 x 12 = 300.

6. 12x^2 = 300 ; x = 5