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andrearamos12oxdie5
08.12.2020 •
Mathematics
A hemisphere of radius 3 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.
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Ответ:
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)
Ответ:
45
Step-by-step explanation:
9x60=540
540/12= 45