Find the volume of the largest right circular cone that can be inscribed in a sphere of radius R.

Solution:

Volume is basically the measure of the capacity that a body holds. Various three-dimensional figures have various formulae of volumes.

Let's find the volume of the largest right circular cone in a sphere of radius r.

Let us consider the right circular cone, inscribed in a sphere.

The figure is given below:

The radius of the right circular cone is 'r'

O is the center of the sphere.

'x' is the distance from the base of the cone to the center O.

Let V be the volume of the sphere

The volume of a cone (V) = (1/3)πr²h 

Also, the height of a cone is given in the figure is AC = AO + OC = R + x,

Since it is a right circular cone then, r² = R² - x² by Pythagoras theorem.

Put the value of height h = (R + x) and r² = (R² - x²) in the volume of a cone formula, (V) = (1/3)πr²h 

V = (1/3) × π × (R² - x²) (R + x) --- (eq 1)

= (1/3) × π × (R³ + R²x - Rx² - x³ ) 

Now, differentiate the volume V w.r.t. x

dV/dx = d[(1/3)π( R³ + R²x - Rx² - x³ )]/dx

= (π/3)[0 + R² - 2Rx - 3x²]

= (π/3)[R² - 2Rx - 3x²]

In order to find the volume, double differentiate V w.r.t. x,

d²V/ dx² = π/3[-2R - 6x]

For a maximum or minimum value of V,

dV/dx = 0

R² - 2Rx - 3x² = 0

Solving the above equation, we have

⇒(R + x)(R - 3x) = 0

⇒ x = -R, R/3

But x ≠ -R (As a length R cannot be negative)

So, when x = R/3 , d²V/dx² = π/3[-2R - 6R/3] = π/3[-2R - 2R] = -4Rπ/3 < 0 

⇒ Volume is maximum only when x = R/3

Substitute x = R/3 in (eq 1)

Maximum Volume of Cone in a Sphere of radius R = (π/3) [R² - R²/9][ R + R/3]

= (π/3) [8R²/9][4R/3]

= (32/81)πR³

= (8/27) [(4/3)πR³) or

= (8/27) × (volume of sphere)

Therefore, the volume of the largest right circular cone that can be inscribed in a sphere of the radius R is (32/81)πR³ cubic units or (8/27) times the volume of the sphere.

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Find the volume of the largest right circular cone that can be inscribed in a sphere of radius R.

Summary:

The volume of the largest right circular cone that can be inscribed in a sphere of the radius R is (32/81)πR³ cubic units or (8/27) times the volume of the sphere.