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.

Answer: Tank dimensions:

s = 12 ft

h = 6 ft

Step-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) times

h = the height of the tank

V = 864 ft³ V = s² * h ⇒ h = V/s² ⇒ h = 864 / s²

Then we have:

A (total) = Base area + 4 * lateral area

Area of the base is s²

Lateral area is s * h but we have 4 wall so lateral area = 4*s*h

Then Objective function is:

A (s) = s² + 4*s * V/s² ⇒ A (s) = s² + 4*864/s ⇒ A (s) = s² + 3456/s

Taken derivative of the objective function:

A' (s) = 2s - 3456 / s²

Solving for s

2s - 3456/s² = 0 ⇒ 2*s³ - 3456 = 0 ⇒ s³ = 3456/2 ⇒ s³ = 1728

s = ∛1728 ⇒ s = 12 ft

So h = V/s² ⇒ h = 864 / 144 ⇒ h = 6 ft