A hemisphere of radius 6 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the cylinder of maximum volume.
find the max radius and height.

x =  4,24 units    radius of the base of the cylinder

h = 4,24  units     height f the cylinder

Step-by-step explanation: See Annex

Let call  " x "  and  " h "  radius and height f th cylinder, then

Volume of the cylinder is :

V(c)  = π*x²*h       (1)

From annex we can see from the right triangle

L²   =  h²  +  x²

As  L  is radius of the hemisphere, and is equal to 6

(6)²  =  h²  +  x²    ⇒   h²  =  36  -  x²   ⇒  h  =  √ (36  -  x²)

Plugging the value of h  in equation (1) we get Volume as a function of x

V(c)  = π*x²*h  

V(x)  = π*x²* √ (36  -  x²)

Taking derivatives :

V´(x)  =  2*π*x*√ (36  -  x²) +(  π*x²) * (-2*x) / √ (36  -  x²)

V´(x) = 0

2*π*x*√ (36  -  x²) - 2*π*x³/ √ (36  -  x²)  =  0

2*π*x*( 36  -  x² )  -  2*π*x³  = 0

72*π*x -  2*π*x³ - 2*π*x³  = 0

18 -  x²  =  0

x = √ 18

x =  4,24 units

And h  =  √ 36 - 18

h  = √18

h = 4,24 units

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