A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in by 12 in by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x.
after cutting two tabs out, the sides are 12-2x
v(x) = x(12-2x)^2
To find the volume, we need to determine the dimensions of the box when it is folded up.
Since we are cutting out squares of side x at each corner, the dimensions of the resulting base of the box will be (12 - 2x) inches by (12 - 2x) inches.
The height of the box will be x inches since we are folding up the sides.
Therefore, the volume of the box, V, can be expressed as:
V = (base area) × height
V = (12 - 2x) × (12 - 2x) × x
V = (144 - 48x - 48x + 4x^2) × x
V = (4x^2 - 96x + 144) × x
V = 4x^3 - 96x^2 + 144x
So, the volume of the box can be expressed as a function of x as V(x) = 4x^3 - 96x^2 + 144x.
To find the volume of the box, we need to determine its dimensions in terms of x.
When squares of side length x are cut out from each corner of the 12 in by 12 in cardboard, the dimensions of the resulting rectangle will be:
Length = 12 in - 2x
Width = 12 in - 2x
To find the height of the box, we observe that it is the same as the side length, x, of the squares that were cut out.
Therefore, the volume of the box, V, can be calculated by multiplying the length, width, and height:
V = (12 in - 2x)(12 in - 2x)(x)
Simplifying the expression:
V = (144 - 24x + 4x^2)(x)
Therefore, the volume of the box, V, can be expressed as a function of x as:
V(x) = x(144 - 24x + 4x^2)