A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 18 in. by 30 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.
nevermind i figured this one out. the answer I got for this one is x(18-2x)(30-2x).
4x^3 -96x^2+540x
To find the volume of the box, we need to determine the dimensions of the box after folding up the sides.
If we remove squares of side x from each corner, then the length of the box will be (30 - 2x) inches, and the width will be (18 - 2x) inches. The height of the box will be simply x inches.
Therefore, the volume of the box can be calculated by multiplying the length, width, and height:
V = (30 - 2x)(18 - 2x)(x)
We can simplify this by expanding the expression:
V = (540 - 60x - 36x + 4x^2)(x)
V = (540 - 96x + 4x^2)(x)
V = 4x^3 - 96x^2 + 540x
Hence, the volume of the box, V, can be expressed as a function of x as V = 4x^3 - 96x^2 + 540x.