Q14. A box with an open top is formed by cutting squares out of the corners of a rectangular piece of cardboard and then folding up the sides. If x represents the length of the side of the square cut from each corner, and if the original piece of cardboard is 19 inches by 16 inches, what size square must be cut if the volume of the box is to be 352 cubic inches?

Background

Topic: Quadratic and Polynomial Applications (Optimization)

This problem involves maximizing or finding a specific volume for a box created by cutting equal squares from each corner of a rectangular piece of cardboard and folding up the sides. It tests your ability to model a real-world scenario with a polynomial equation and solve for a variable.

Key Terms and Formulas

• Volume of a box:

• Let be the side length of the square cut from each corner.

• After cutting and folding, the dimensions become:

  • Length:

  • Width:

  • Height:

• Volume equation:

Step-by-Step Guidance

1. Write the equation for the volume of the box in terms of :

2. Set the volume equal to 352 cubic inches:

3. Expand the expression to get a quadratic in terms of :

   First, multiply out the binomials:

4. Simplify the expanded expression and substitute back into the volume equation. You should now have a cubic equation in .

5. Rearrange the equation so that all terms are on one side, setting the equation equal to zero. At this point, you can use factoring, the Rational Root Theorem, or another method to solve for .

Try solving on your own before revealing the answer!