Q15. An open box is to be made from a square sheet of cardboard by cutting out 4-inch squares from each corner as shown and then folding along the dotted lines. If the box is to have a volume of 400 cubic inches, find the original dimensions of the sheet of cardboard.

Background

Topic: Quadratic and Cubic Equations in Geometry (Applications)

This problem involves modeling a real-world scenario with a cubic equation. You are asked to relate the dimensions of a box formed by cutting and folding a square piece of cardboard to its volume, and then solve for the original size of the cardboard.

Key Terms and Formulas

• Volume of a box:

• Original side length of cardboard: Let be the side length of the original square sheet (in inches).

• Cut size: 4 inches from each corner.

• After cutting and folding: The height of the box is 4 in, and the base is in by $(x - 8)$ in.

Step-by-Step Guidance

1. Let be the original side length of the square sheet of cardboard (in inches).

2. After cutting 4-inch squares from each corner, the dimensions of the base become inches by $(x - 8)$ inches, and the height is 4 inches.

3. Write the equation for the volume of the box: .

4. Set the volume equal to 400 cubic inches: .

5. Divide both sides by 4 to isolate .

Try solving on your own before revealing the answer!