A box with a square base and no top is to be made from a square piece of carboard by cutting 5 in. squares from each corner and folding up the sides. The box is to hold 23805 in3. How big a piece of cardboard is needed?
If the cardboard has side length x, then
5(x-10)^2 = 23805
Now just solve for x.
To find out the size of the cardboard needed to make the box, we need to determine the dimensions of the square base first.
Let's assume that the length of each side of the square base is x inches.
Since we are cutting 5-inch squares from each corner, the length of the base after folding up the sides will be reduced by 10 inches (5 inches from each side).
Therefore, the length of the square base of the box will be (x - 10) inches.
The height of the box is determined by folding up the sides, which will be 5 inches.
So, the volume of the box can be calculated by multiplying the length, width, and height:
Volume = Length × Width × Height
23805 in³ = (x - 10) in × (x - 10) in × 5 in
Now, let's solve this equation to find the value of x:
23805 in³ = (x - 10) in × (x - 10) in × 5 in
23805 in³ = 5 in × (x - 10) in × (x - 10) in
4,761 in³ = (x - 10) in × (x - 10) in
Taking the square root of both sides:
√4,761 in³ = √((x - 10) in × (x - 10) in)
√4,761 in = x - 10 in
Simplifying:
69 in ≈ x - 10 in
Now, let's solve for x:
x ≈ 69 in + 10 in
x ≈ 79 in
Therefore, the length of each side of the square base should be approximately 79 inches.
Since we are using a square piece of cardboard, we need to account for the sides that are folded up. So the size of the cardboard needed will be the length of the base plus the width of the folded sides, which is:
Cardboard Size = Length of Base + 2 * Folded Side Width
Cardboard Size = 79 in + 2 * 5 in
Cardboard Size = 79 in + 10 in
Cardboard Size = 89 in
Hence, a piece of cardboard that is approximately 89 inches square is needed to make the box.
To find out how big a piece of cardboard is needed, we need to consider the dimensions of the box after folding.
Let's assume the length of each side of the square base is "x" inches. Since 5-inch squares are cut from each corner, the dimensions of the base of the box will be reduced by 10 inches (5 inches removed from each side).
So, the length and width of the base of the box will be (x - 10) inches.
When the sides are folded up, they will form the height of the box.
Now let's calculate the volume of the box.
The volume of a rectangular box is given by the formula: volume = length * width * height.
In this case, the length and width of the box's base are (x - 10), and the height will be 5 inches, as the cut-out squares are 5 inches.
So, the volume of the box can be written as:
V = (x - 10) * (x - 10) * 5
Given that the volume of the box needs to be 23805 in³, we can set up the equation:
23805 = (x - 10) * (x - 10) * 5
Now we can solve for x.
Divide both sides of the equation by 5:
4761 = (x - 10) * (x - 10)
Take the square root of both sides of the equation:
√4761 = x - 10
Simplify the square root:
69 = x - 10
Now, isolate x by adding 10 to both sides:
x = 69 + 10
x = 79
Therefore, the length of each side of the square piece of cardboard needed to make the box is 79 inches.