Скачать или смотреть Largest volume of cylinder and cone inscribed in sphere| How to find the largest volume

The largest volume of cylinder and cone inscribed in sphere| How to find the largest volume

The radius of the Sphere is R, the base radius of the Cylinder is r and the height is h. Point O is the center of the Sphere.
Let, OA=x
Apply Pythagoras theorem in △OAC,
x²+r²=R² ………(i)
h=2x
The volume of the Cylinder (V)= 𝜋r²h

V= 𝜋r².(2x)
V=2𝜋.(R²-x²).x
V=2𝜋.(R²x-x³) ………(ii)

Differentiate both sides,
dV/dx=2𝜋.(R²-3x²)
The second-order derivative is negative
So, 2𝜋.(R²-3x²)=0
x=R/√3 ………(iii)

h=2x=2R/√3
r=√(2/3).R
The Maximum Volume of the Cylinder, V= 𝜋r²h
V=(4𝜋/3√3).R³
The volume of the largest Cone inscribed in a Sphere
V=(32/81). 𝜋R³

The ratio between the Volumes of the Cylinder and the Cone= 9√3 :8

Maxima and Minima from Calculus
Let, f(x) be a function of x and y=f(x)
Here, the derivative of the function dy/dx is geometrically represented as the slope of the curve. The slope is positive if the function is increasing, Zero at the maximum or minimum, and negative just after the maximum.

The second derivative d²y/dx² is the rate of change of the derivative. If the second derivative is negative at a critical point, then the critical point is a local maximum. If it’s positive, the critical point is a local minimum.

Volume and Surface area of Sphere:    • Total Surface area and Volume of the spher...  

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Home Work Problem (Will be discussed in the next Video):
Find the sum of the volumes of infinite Spheres inscribed in a right circular cone?

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