WHat is the volume of the box? DOmain of the box

Question

an open box is to be made from a flat piece of material 30 inches long and 6 inches wide by cutting equal squares of length x from the corners and folding up the sides.

What is the volume of the box?

WHat is the domain of the box in interval notation

Answer 1

The length of the box is (30-2x) because you are removing x from both ends of the box

The width of the box is (6-2x) because you are removing x from both ends of the box

the height is x because when you fold up the flaps you've created they will come x high

Volume of a rectangular prism is length x width x height

V=lwh

plug in values

V=(30-2x)(6-2x)x

Expand if needed:

V=(180-60x-12x+4x²)

V=180x-72x²+4x³

Domain is the possible values for x. Since this is a real 3D shape, l,w,h cannot be negative. The smallest x could be and not result in a negative value is x=0: l=30, w=6, h=0 although this wouldn't create a box just a flat piece of cardboard.

The largest would be what would make either w=0 or l=0

6-2x=0

6=2x

3=x

30-2x=0

30=2x

15=x

Since x=3 is smaller, this is the largest the x value can be since any bigger width would be negative.

Domain would be:

[0,3] <--- if you count a zero volume box to be valid

(0,3) <--- if you only count a box where V>0

Answer 2

The 30" x 6" piece of cardboard is shown in the middle.

Think about cutting out each square at the corner and folding up the remaining "flaps".

This gives you the box on the bottom.

The length is the length of the cardboard with a square missing from both sides, so 30-2x

The width is the width of the cardboard with a square missing from both sides, so 6-2x

The height is the length of each flap that was folded up, and that is x

So putting it all together, V = l w h. V = (30-2x)(6-2x)(x) which multiplies out to

V(x) = 4x³-72x²+180x

But we cannot use just any x. We need to keep the length and the width from becoming 0, so

we need to solve the inequalities 30-2x>0 and 6-2x>0

These give us x<15 and x<3 -- the most restrictive is that x<3, so that is the domain.