I'm a little confused as to how to solve this word problem I have. The problem is: A rectangular box (with a top) has a square base. The sum of the lengths of its edges is 8 feet. What dimensions should the box have in order for its surface area to be as large as possible?
My work so far is:
x = Length of side of base (ft)
y = Height of rectangular box (ft)
M = Surface area of box (ft2)
The question asks for the largest surface area possible with a rectangular box that has a square base and the sum of whose edges is 8 ft. To determine that, we must first determine the formulas with the given information for the sum of the edges and the surface area. The area of a square with side length x is x2, and the area of a rectangle with side lengths x and y is xy.
4y + 8x = 8
4y = 8 – 8x
y = 2 – 2x
M = 2x2 + 4xy
M = 2x2 + 4x(2-2x)
M = -6x2 + 8x
And that is the quadratic equation for the surface area
Also, for the bottom line, if that is truly the equation, would that be considered the equation for the surface area, or should I reword it to something else? Thanks!
You want to maximize the surface area, so
$$M(x) = -6x^2 + 8x \implies M'(x) = -12x + 8$$
Now, $$M'(x) = 0 \implies -12x + 8 = 0 \implies x = \frac{8}{12} = \frac{2}{3}$$
So your surface area is a maximum when $x = \frac{2}{3}$ and thus $y = 2\left(1-\frac{2}{3}\right) = \frac{2}{3}$
So essentially you want a cube.