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

x = 0.629 cm

Step-by-step 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