A rectangular tank with a square base, an open top, and a volume of 864 ft cubed is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area. Let s be the length of one of the sides of the square base and let A be the surface area of the tank. Write the objective function.
Accepted Solution
A:
Answer: Tank dimensions:s = 12 fth = 6 ftStep-by-step explanation:s = lenght side of the square base A the total area of of the tank ( this area is the area of the base plus the lateral area (4) timesh = the height of the tankV = 864 ft³ V = s² * h ⇒ h = V/s² ⇒ h = 864/ s²Then we have : A (total) = Base area + 4 * lateral area Area of the base is s²Lateral area is s * h but we have 4 wall so lateral area = 4*s*hThen Objective function is :A(s) = s² + 4*s * V/s² ⇒ A(s) = s² + 4*864/s ⇒ A(s) = s² + 3456/sTaken derivative of the objective function:A´(s) = 2s - 3456/ s²Solving for s2s - 3456/s² = 0 ⇒ 2*s³ - 3456 = 0 ⇒ s³ = 3456/2 ⇒ s³ = 1728s = ∛1728 ⇒ s = 12 ftSo h = V/s² ⇒ h = 864 / 144 ⇒ h = 6 ft