Unit 2 Exam Review – Applications and Problem Solving in College Algebra

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: $V = \text{length} \times \text{width} \times \text{height}$

• Original side length of cardboard: Let $x$ 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 $(x - 8)$ in by $(x - 8)$ in.

Step-by-Step Guidance

1. Let $x$ 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 $(x - 8)$ inches by $(x - 8)$ inches, and the height is 4 inches.

3. Write the equation for the volume of the box: $V = (x - 8)(x - 8)(4)$.

4. Set the volume equal to 400 cubic inches: $(x - 8)^2 \times 4 = 400$.

5. Divide both sides by 4 to isolate $(x - 8)^2$.

Try solving on your own before revealing the answer!

Final Answer: The original dimensions of the sheet are 18 inches by 18 inches.

Solving $(x - 8)^2 = 100$ gives $x - 8 = 10$, so $x = 18$ inches.

This means the original square sheet must be 18 inches on each side to create a box with the required volume.