Answer:
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
