Open-top box. Thomas is going to make an open-top box

by cutting equal squares from the four corners of an
11 inch by 14 inch sheet of cardboard and folding up the
sides. If the area of the base is to be 80 square inches, then
what size square should be cut from each corner?

The parts cut out (x) have to be square or else the heights of the box won't match.

The area of the base is to be 80 square inches.
The lengths and sides are 11 and 14, therefore the dimensions of the base would be (11-2x) and (14-2x).
We get the constraint / relationship:
(11-2x)(14-2x) = 80
Solve the quadratic equation for x.
Unfortunately the equation does not factor, so you will need the quadratic formula to solve it. It should be between 1.5 and 2.

To find the size of the square that should be cut from each corner, we need to follow these steps:

1. Start by visualizing the situation. The sheet of cardboard has dimensions 11 inches by 14 inches. Thomas will cut squares from each corner and fold up the sides to create an open-top box.

2. Let's assume that Thomas cuts squares with side length x inches from each corner. As a result, the length of the box's base will be reduced by 2x inches, and the width will be reduced by 2x inches.

3. The new dimensions of the box will be (11 - 2x) inches by (14 - 2x) inches.

4. The area of the base of the box is given as 80 square inches. We know that the area of a rectangle is calculated by multiplying its length by its width. So, we can set up the equation:

(11 - 2x) * (14 - 2x) = 80

5. Simplify and solve the equation. Expand the equation:

154 - 22x - 28x + 4x^2 = 80

Combine like terms:

4x^2 - 50x + 154 = 80

Subtract 80 from both sides:

4x^2 - 50x + 74 = 0

6. Now, we have a quadratic equation. To solve it, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 4, b = -50, and c = 74. Let's substitute these values:

x = (50 ± √((-50)^2 - 4 * 4 * 74)) / (2 * 4)

Simplify the square root:

x = (50 ± √(2500 - 1184)) / 8

x = (50 ± √(1316)) / 8

x = (50 ± 36.271) / 8

7. Now, we have two possible solutions for x: (50 + 36.271) / 8 and (50 - 36.271) / 8. Calculate these values:

x ≈ 8.284 or x ≈ 2.715

8. Since we are dealing with a physical object, x cannot be negative or larger than the dimensions of the cardboard. Therefore, we discard x ≈ 8.284 as it's too large for the given dimensions.

9. Finally, the appropriate size square that should be cut from each corner is approximately 2.715 inches.