x = width of the base.
2x = length of the base.
h = height of the base.
Area of the base is x * 2x = 2x^2
Area of each long side is 2xh. There are two long sides, so total is 4xh
Area of each short side is xh. There are two long sides, so total is 2xh.
The total area of the sides is 2xh + 4xh = 6xh
Volume = area of base * h = 2x^2 * h = 2hx^2 = 28
Solve for h: h = 28/2x^2 = 14/x^2
Cost is 10* area of base + 9* area of sides = 10*2x^2 + 9*6xh = 20x^2 + 9*6x*14/x^2 = 20x^2 + 756x/x^2 = 20x^2 + 756/x.
Find the derivative, set = 0. If you graph the original function, you'll see that's a minimum.
Derivative = 40x - 756/x^2 = 0
40x^3 -756 = 0
x^3 = 756/40 = 378/20 = 189/10 = 18.9
x = cuberoot(18.9) is the minimum.
Cost = 20*(18.9)^(2/3) + 756/cuberoot(18.9) = about 425.72
