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

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step-by-step explanation:

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