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A box with no top is to be constructed from a piece of cardboard whose length measures 14 in. more than its width. The box is formed by cutting squares that measure 4 in. on each side from the four corners and then folding up the sides. If the volume of the box will be 288 in³, what are the dimensions of the piece of cardboard? a) Length: 18 in, Width: 10 in b) Length: 16 in, Width: 12 in c) Length: 12 in, Width: 14 in d) Length: 20 in, Width: 8 in

User Wilfrido
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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:

User Aristoteles
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