From each corner of a square piece of cardboard, remove a square of sides 3 inch. Turn up the edges to form an open box. If the box is to hold 300 inch cubed, what are the dimensions of the original piece of cardboard?
v = 3(x-6)^2
when v=300, we have (x-6)^2 = 100, so x = 16.
Well, it looks like the square cardboard piece had four corners. Now, if we remove a square piece from each corner, what will be left is an open box made by turning up the edges. I hope you brought your imagination along for this ride!
Now, since the box needs to hold 300 inches cubed, we know that the volume of the box is 300 cubic inches.
To find the dimensions of the original piece of cardboard, let's start with the length and width of the square, which we'll call "x".
When we remove the squares from each corner, the length of the box will be (x-6) inches and the width will be (x-6) inches as well. Why minus 6? Well, each side of the square we removed has a length of 3 inches, so we have 3 inches times 2 (for the length and width) which gives us a total of 6 inches.
Now, to find the height of the box, we need to consider the fact that when we fold up the edges, they will form the height. Since the original square cardboard piece had no height, we can say that the height is also 0 inches.
To find the volume of the box, we multiply the length, width, and height. So, (x-6)(x-6)(0) = 300.
Well, it seems like we have a bit of a problem here. Since the height is 0, the volume of the box will always be 0. Therefore, it's impossible for the box to hold 300 inches cubed.
Looks like someone made a square mistake!
To solve this problem, we need to follow these steps:
1. Visualize the problem: Draw a square on a piece of paper to represent the original cardboard. Label the side of the square as "x" since we don't know its length yet.
2. Remove squares from each corner: From each corner of the square, remove a smaller square with a side length of 3 inches. This will leave us with four square flaps on the edges.
3. Determine the dimensions of the box: Fold up the flaps along the edges to form an open box. The base of the box will be a square with sides (x - 6) inches (since we removed two 3-inch squares from each side). The height of the box will be 3 inches.
4. Calculate the volume of the box: Multiply the length, width, and height of the box to find its volume. In this case, we have (x - 6) * (x - 6) * 3 = 300 cubic inches.
5. Solve the equation for x: Simplify the equation and solve for x.
- Multiply (x - 6) by itself: (x - 6) * (x - 6) * 3 = 300
- Expand the equation: 3(x^2 - 12x + 36) = 300
- Distribute the 3: 3x^2 - 36x + 108 = 300
- Subtract 300 from both sides: 3x^2 - 36x - 192 = 0
- Divide everything by 3: x^2 - 12x - 64 = 0
- Factor the quadratic equation: (x - 16)(x + 4) = 0
- Set each factor equal to zero: x - 16 = 0 or x + 4 = 0
- Solve for x: x = 16 or x = -4
6. Check if the solution is valid: Since the length of the cardboard cannot be negative, we discard x = -4. Therefore, the length of the original piece of cardboard is x = 16 inches.
So, the dimensions of the original piece of cardboard are 16 inches by 16 inches.