Question

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 6 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to three decimal places.) g

Answer 1

x = 0.629 cm

Explanation:

The volume of the solid is:

V(s) = V(c) + V ( two hemisphere)

And V(s) = 6 cm³

The volume of the cylinder is V(c) = π*x²*h

Let call " x " the radius of the base f the cylinder and of course the radius of the hemispheres

The volume of the cylinder is V(c) = π*x²*h

And is equal to 6 - Volume of the sphere of radius x ( volume of two hemisphere of the same radius is just one sphere

Then V(c) = 6 - (4/3)*π*x³

Then V(c) = π*x²*h = 6 - (4/3)*π*x³

h = [ 6 - (4/3)*π*x³ ] /π*x²

The lateral area of the cylnder is :

A(l) = 2*π*x * h ⇒ A(l) = 2*π*x * [ 6 - (4/3)*π*x³ ] /π*x²

A(l) = 12/x - (8/3)*π*x²

Then surface of the area of the cylinder is:

S(c) = A(b) + A(l) ⇒ S(c) = π*x² + 12/x - (8/3)*π*x²

And the area of a sphere is

S(sphere) = 4π*x²

Total area of the solid is:

S(s) = π*x² + 12/x - (8/3)*π*x² + 4π*x²⇒ S(s) = 5*π*x²+ 12/x - (8/3)*π*x²

Taking derivatives on both sides of the equation we get

S´(s) = 10*π*x - 12/x² - (16/3)*π*x

As 10 = 30/3

S´(s) = (46/3)*π*x - 12/x²

S´(s) = 0 (46/3)*π*x - 12/x² = 0

46*π*x³ = 36

x³ = 0,2492

x = ∛0,2492

x = 0.629 cm