The frame for a shipping crate is to be constructed from 24feet of 2x2 lumber.

If the crate is to have square ends of side x feet, express the outer volume v of the crate as a function of x (disregard the thickness of he lumber).

Well, I know that v=lwh, but I don't know what to do with this.

done by Steve

http://www.jiskha.com/display.cgi?id=1349250036

To express the outer volume of the crate (v) as a function of x, we need to determine the dimensions of the crate in terms of x.

A shipping crate has six rectangular faces (top, bottom, and four sides). Since the crate has square ends of side x feet, the top and bottom faces will also be squares of side x.

The remaining four faces form the sides of the crate. Since these are rectangles, their dimensions will depend on x.

To determine the dimensions of the sides, let's consider one side. We know that the crate has 24 feet of 2x2 lumber, which means the total length of the lumber used for all four sides is 24 feet. Each side piece consists of two 2x2 lumber pieces, so the total length of the sides is equal to 24/2 = 12 feet.

Since there are four sides, each side will have a length of 12/4 = 3 feet. The sides will also have a height equal to the height of the square ends, which is x.

Therefore, the dimensions of the crate will be as follows:
Top and bottom face: x by x (square)
Sides: x by 3 (rectangles)

To calculate the outer volume (v) of the crate, we can use the formula v = lwh. In this case, the length (l) will be equal to the sum of the lengths of the four sides.

The length of the four sides is equal to 2 * (x + 3) since each side has a length of x and there are two opposite sides.

Now we can express the outer volume (v) of the crate as a function of x:

v = (x * x) + 2 * (x + 3) * x

Simplifying this equation will give you the final expression for the outer volume of the crate as a function of x.