An open box is to be made from a square piece of cardboard, 32 inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure below). Determine the function, V, in terms of x, that represents the volume of the box.
A. -2x^3+32x^2
B.-4x^3+64x^2
C.4x^3-64x^2+32x
D.-4X^3+64x^2-32x
E.4x^3-128x^2-1024x
volume = x(32-2x)^2
which of your answers matches the above once you expand it?
Idk
To determine the function that represents the volume of the box in terms of x, let's first find the height of the box when the sides are folded up.
When you cut squares with sides of length x from each corner of the 32-inch square piece of cardboard, the resulting height will be x. This is because when you fold up the sides, the cut corners will form the bottom of the box, and the height will be equal to the length of the cut squares.
The dimensions of the base of the box will be (32 - 2x) inches by (32 - 2x) inches, since each side loses 2x inches due to the cut squares. Therefore, the area of the base will be (32 - 2x) * (32 - 2x) square inches.
Now, to find the volume of the box, we multiply the height (x inches) by the area of the base.
V(x) = x * (32 - 2x) * (32 - 2x)
Simplifying the expression:
V(x) = (32 - 2x)^2 * x
= (32 - 2x)(32 - 2x) * x
= (1024 - 128x + 4x^2) * x
Expanding:
V(x) = 1024x - 128x^2 + 4x^3
So, the function that represents the volume of the box in terms of x is:
V(x) = 4x^3 - 128x^2 + 1024x
Therefore, the correct answer is: E. 4x^3 - 128x^2 + 1024x.
To determine the volume of the box, we need to calculate the product of its length, width, and height.
The length and width of the box will be the dimensions of the base, which is formed by the original square cardboard minus the squares cut from each corner. After cutting squares with sides of length x from each corner, the length and width of the base will be reduced by 2x. Therefore, the length of the base will be 32 - 2x, and the width of the base will also be 32 - 2x.
The height of the box will be equal to the length of the squares cut from the corners, which is x.
Now, we can express the function, V, that represents the volume of the box:
V = (32 - 2x) * (32 - 2x) * x
V = (32 - 2x)^2 * x
Expanding the square:
V = (1024 - 128x + 4x^2) * x
V = 1024x - 128x^2 + 4x^3
Therefore, the function, V, that represents the volume of the box in terms of x is 4x^3 - 128x^2 + 1024x.
So, the correct answer is E. 4x^3 - 128x^2 - 1024x.