Equal squares of side length x are removed from each corner of a 20 inch by 30 inch piece of cardboard, and the sides are turned up to form a box with no top. Write the volume V of the box as a function of x.
V = x (30-2x)(20-2x)
multiply that out
Of course part 2 will be to take the derivative and set it to zero to find the height for maximum volume.
To find the volume V of the box, we need to determine its dimensions and then calculate the product of its length, width, and height.
When equal squares of side length x are removed from each corner of a rectangular piece of cardboard, the length and width of the piece of cardboard decrease by 2x.
Therefore, the length of the box will be (30 - 2x) inches, and the width of the box will be (20 - 2x) inches.
To find the height of the box, we will use the value of x as the height of the removed squares.
Therefore, the volume V of the box can be calculated as follows:
V = (length) * (width) * (height)
= (30 - 2x) * (20 - 2x) * x
= (600 - 120x - 40x + 4x^2) * x
= (4x^2 - 160x + 600) * x
= 4x^3 - 160x^2 + 600x
So, the volume V of the box is given by the function V(x) = 4x^3 - 160x^2 + 600x.
To find the volume V of the box, we need to determine the height, length, and width of the box.
First, let's determine the height of the box. When the corners are cut and folded up, the height of the box will be equal to the side length of the squares that were removed, which is x.
Next, let's determine the length of the box. After cutting off equal squares of side length x from each corner, the remaining length of the cardboard will be decreased by 2x on each side. Therefore, the length of the box will be (30 - 2x) inches.
Similarly, the width of the box will be (20 - 2x) inches.
Now, we can calculate the volume V of the box by multiplying the height, length, and width:
V = x * (30 - 2x) * (20 - 2x)
Therefore, the volume V of the box can be expressed as a function of x: V(x) = x * (30 - 2x) * (20 - 2x).