Respuesta :
Answer:
R = √(2/3)
h = 1/√3
Step-by-step explanation:
Given that
The volume of a cylinder, V = πr²h
Assuming that the hypotenuse of a right angles triangle in the cylinder is 1. This means the square of the adjacent and the opposite sides is equal to one. And thus,
r² + h² = 1
r² = 1 - h²
If you substitute for r² in the volume of the cylinder, you have
V = π(1 - h²)h
V = πh - πh³
To find the critical point, we differentiate V, so that
dV = π - 3πh², if we equate this to zero, we have
π - 3πh² = 0
π = 3πh²
1 = 3h²
h² = 1/3
h = 1/√3
Also, to find for which value of h do we have maximum or minimum volume, we differentiate again. So that.
V'' = 0 - 6πh
On substituting for h, we have
V'' = -6π * 1/√3
This invariably means that V'' of 1/√3 is less than 0(since it's negative)
Going back to the first formula of r, we have
r² = 1 - h²
On substituting for h, we have
r² = 1 - (1/√3)²
r² = 1 - 1/3
r² = 2/3
r = √(⅔)
The radius and height of the cylinder of maximum volume are;
Radius = √6
Height = √3
Formula for volume of a cylinder is;
V = πr²h
We are told that the hemisphere has a radius of 3 and sits on a horizontal plane. This means that the radius of the sphere will be the diagonal gotten from the intersection of the radius (r) of the cylinder and the height (h) of the cylinder. Thus, from Pythagorean theorem, we have;
r² + h² = 3²
r² = 9 - h²
Volume is now;
V = π(9 - h²)h
V = π(9h - h³)
Let us find the derivative of the volume and equate to zero the find the height and radius at maximum volume. Thus;
dV/dh = π(9 - 3h²)
At dV/dh = 0;
π(9 - 3h²) = 0
Thus; 3h² = 9
h² = 9/3
h² = 3
h = √3
Thus; r² = 9 - (√3)²
r² = 9 - 3
r² = 6
r = √6
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