In interval notation (17) 3.2 Polynomial functions

Question

A cardboard box has a square base, with each edge of the base having length x inches, as shown in the following figure. The total length of all 12 edges of the box is 144 in.

A box with a square base is given. Two sides of the base are labeled x.

(a)

Show that the volume of the box is given by the function V(x) = 2 x²(18 − x).

The volume of a rectangular prism is the area of the base times the height. Determine the area of the base in terms of x.

 

Use the given information to write the height of the box in terms of x.

 

Combining the previous results we see that the area of the box given is V(x) = 2 x²(18 − x).

(b)

What is the domain of V? (Use the fact that length and volume must be positive.

 

Answer 1

Tori,

Since the base is a square and the area of a square (or a rectangle, which a square is a type of rectangle) is length times width, (lxW), the area of the base will be x times x, or x².

This leaves us with the heigh of the prism. Let h represent the height. Since the total length of all 12 edges of the box is 144 in, then

4h+8x=144 (4 comes from the four vertical edges, each measuring h inches)

Also there are 8 edges of the squares top and bottom.

using algebra to solve for h (subtract 8x from both sides and then divide by 4), we get the height in terms of x: 36 − 2x.

since area of the square is x² and the height is 36 − 2x, then the volume will be V(x)= x²(36 − 2x), which if we factor a 2 from the quantity (36 − 2x), we end up with the desired V(x) = 2 x²(18 − x).

The domain is a bit tricky. There is no way we can have a negative length, so the smallest x (which represents the length of one side of the base) can be is 0.

To come up with a higher bound, we need to look at how tall the prism can get. Since the height is 36 − 2x, then the largest x can be would be 18, otherwise, we end up with a negative height. A graph is also useful to see this relationship. In interval notation: (0,18), since I think a base of 0 length is a bit silly and if x=18, then h =0, which is also silly.

Answer 2

top and bottom are 2x^2 = Area

bottom Area = x^2

height = (144- 8x)/4=36-2x

V= hA = (36-2x)(x^2)

= 2x^2(18-x)

domain of x is (0, 18)

or 0<x,18