+128406 Volume = L * W * H = 70
But L = 2W
So we have that
2W * W * H = 70
2W^2 * H = 70
W^2 * H = 35.......solve for the height
H = 35/W^2
The surface area is the top and bottom = [ 2 * L * W] = [ 2 *2W*W] = 4W^2
And the total cost per m^2 for this part of the box is $9 * Area = 9*4W^2 = 36W^2
The surface area of two of the sides = 2*2W * H = 2*2W* (35/ W^2) =
140/W
And the surface area for the other two sides = 2*W*H = 2*W (35/W^2) = 70/W
So....the total surface area of the sides = (140/W + 70/W) = 210/W
And the total cost per m^2 of this part of the box is $6 * Area = 6*210/W = 1260/W = 1260W^(-1)
So....the total cost, C, is given by
C = 36W^2 + 1260W^(-1) take the derivative and set to 0
C' = 72W - 1260W^(-2) = 0
C' = 72W - 1260/W^2 = 0
72W = 1260/W^2
72W^3 = 1260
W^3 = 17.5
W = 3√17.5 m
L = 2W = 2 3√17.5 m
H = 35 /W^2 = 35 / 3√(17.5)^2 = 35 / 3√306.25 m
The minimum cost is
36 W^2 + 1260 / W =
36 3√[17.5^2] + 1260 / 3√17.5 =
36 3√306.25 + 1260 / 3√17.5 ≈ $727.97
