Answer:
To find the dimensions of the piece of cardboard, you can follow these steps:
1. Let's assume that the width of the piece of cardboard is "x" inches. Since the length is 14 inches more than the width, the length will be "x + 14" inches.
2. To create the box, you will cut squares with a side length of 4 inches from each corner of the cardboard.
3. When you cut out these squares, you will reduce both the length and width of the cardboard by 8 inches (4 inches from each corner).
4. After cutting out the squares, the dimensions of the cardboard will be:
- Length = (x + 14) - 2(4) = x + 6 inches
- Width = x - 2(4) = x - 8 inches
5. Now, you can calculate the volume of the box. The volume of a box is given by the formula: Volume = Length × Width × Height.
6. In this case, the height of the box is 4 inches (because you folded up the sides of the cardboard), and you want the volume to be 288 in³. So, you have the equation:
288 = (x + 6)(x - 8)(4)
7. Simplify the equation:
288 = 4(x + 6)(x - 8)
8. Divide both sides by 4:
72 = (x + 6)(x - 8)
9. Now, expand and rewrite the equation as a quadratic equation:
x² - 2x - 48 = 0
10. Factor the quadratic equation:
(x - 12)(x + 4) = 0
11. Set each factor equal to zero and solve for x:
x - 12 = 0 or x + 4 = 0
12. Solve for x in both equations:
x = 12 or x = -4
Since the width of the cardboard cannot be negative, we take x = 12 inches as the width.
Now, we can find the length:
Length = x + 14 = 12 + 14 = 26 inches
So, the dimensions of the piece of cardboard are:
Width: 12 inches
Length: 26 inches
The correct answer is not among the options provided. It seems there might be a mistake in the answer choices.
Explanation: