the frame for a shipping crate is to be constructed from 24 feet of 2 by 2 lumber.
if the crate is to have square ends of side x feet, express the outer volume V as a function of x (disregard the thickness of the lumber)
V = x^2 * y
where 8x + 4y = 24
so, y = 6-2x
V = x^2 (6-2x)
To express the outer volume, let's first calculate the dimensions of the shipping crate.
Since the crate has square ends of side x feet, the length, width, and height of the crate will also be x.
Now, let's find the dimensions of the crate frame using the given information that it will be constructed from 24 feet of 2 by 2 lumber.
Each side of the lumber is 2 feet and we have 24 feet in total. So, the number of pieces of lumber we have is 24/2 = 12.
To construct the frame, we have four sides, so we need to divide the total length of lumber (12 pieces) equally among the four sides.
Thus, each side of the frame will have 12/4 = 3 pieces of lumber.
Now, let's calculate the dimensions of the frame:
Since each side of the frame has 3 pieces of lumber, the length of the frame will be x + 2 * 3 = x + 6.
The width of the frame will be x + 2 * 3 = x + 6.
The height of the frame will be x.
To calculate the outer volume V as a function of x, we multiply the length, width, and height of the frame:
V = (x + 6) * (x + 6) * x
V = (x + 6)^2 * x
Therefore, the outer volume V of the crate as a function of x is V = (x + 6)^2 * x.