A box with a square base and no top is to be made from a square piece of cardboard by cutting 4-in. squares from each corner and folding up the sides, as shown in the figure. The box is to hold 324 in3. How big a piece of cardboard is needed?
To find out how big a piece of cardboard is needed, we need to determine the dimensions of the square base.
Let's assume that the side length of the square base is 'x'.
When we cut a 4-in. square from each corner, the resulting dimensions of the base will be (x-8) inches by (x-8) inches.
We can calculate the volume of the box using the formula: Volume = length × width × height.
Given that the volume of the box is 324 in^3, we can set up the equation:
324 = (x-8) × (x-8) × 4.
Now let's solve this equation step by step:
1. Expand the equation:
324 = (x^2 - 16x + 64) × 4.
2. Distribute the 4:
324 = 4x^2 - 64x + 256.
3. Subtract 324 from both sides of the equation to set it equal to zero:
4x^2 - 64x + 256 - 324 = 0.
4. Simplify the equation:
4x^2 - 64x - 68 = 0.
5. Divide the entire equation by 4 to simplify it further:
x^2 - 16x - 17 = 0.
6. Factor the quadratic equation:
(x - 17)(x + 1) = 0.
7. Set each factor equal to zero and solve for x:
x - 17 = 0 or x + 1 = 0.
- x - 17 = 0:
- x = 17.
- This is not a valid solution since the side length of the square cannot be negative.
- x + 1 = 0:
- x = -1.
- Again, this is not a valid solution since the side length of the square cannot be negative.
Since neither solution is valid, we can conclude that there is no solution for this equation. Therefore, it is not possible to make a box with a volume of 324 in^3 using a square piece of cardboard without any top.
To find out how big a piece of cardboard is needed, we need to determine the dimensions of the square base and the height of the box.
Let's start by finding the dimensions of the square base. We know that each side of the base will have a length equal to the original side length minus twice the length of the corners that have been cut off.
Let's assume the side length of the original square piece of cardboard is x inches. Since 4-inch squares are cut off from each corner, the resulting side length of the square base will be (x - 2 * 4) inches or (x - 8) inches.
Next, we need to determine the height of the box. We know that the volume of a rectangular box is given by the formula: volume = length * width * height.
In this case, the length and width of the base are the same since it's a square, so both are (x - 8) inches. The height of the box will be the amount of cardboard that we fold up to create the sides.
Given that the volume is 324 cubic inches, we can set up the equation:
324 = (x - 8) * (x - 8) * h,
where h represents the height of the box.
Simplifying the equation:
324 = (x - 8)^2 * h.
Now, we need to solve this equation to find the value of x, which represents the side length of the original square piece of cardboard.
To do this, we can rearrange the equation and solve for x:
324 / h = (x - 8)^2.
Taking the square root of both sides:
√(324 / h) = x - 8.
Adding 8 to both sides:
√(324 / h) + 8 = x.
Now, we have the value of x in terms of h. Since we don't know the height of the box, we can't calculate the exact value of x yet.
To find out how big a piece of cardboard is needed, you would need to specify the height of the box. Once you know the height, you can substitute it into the equation above to calculate the value of x, which represents the side length of the original square piece of cardboard.
If the cardboard has side x, the volume is
(x-8)^2*4 = 324
Now it is clear what x is.