Respuesta :
Answer:
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
