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)
