Question

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box can be made. If the cardboard is 18 in. long and 12 in. wide and the square cutaways have dimensions of x in. by x in., find a function giving the volume V(x) of the resulting box.

Answer 1

Answer:

$V(x)=4x^3-60x^2+216x$

Step-by-step explanation:

We have the situation better explained on figure 1.

The volume of a rectangular prism is V = Area of the base x heigth

In our case, the area of the base is length x width.

Observing  the length of the cardboard we have:

cardboard length = 18 = x + length of the base + x = 2x +length

Solving for the length of the base,

The length = 18 - 2x

The same process is made for the width of the box base:

width of the cardboard = 12 = x + width of the bases + x = 2x + width

Solving for the width of the base,

Width = 12 - 2x

The height for the box will be x.

The volume is V = length x width x height

Replacing the dimensions in terms of x we have,

$V(x) = (18 - 2x)(12 - 2x)(x)=(4x^2-60x+216)x=4x^3-60x^2+216x$

Finally,

$V(x)=4x^3-60x^2+216x$