Find the Maximum Volume of a Cylinder That Can Be Inscribed in a Sphere of Radius 10 Cm in Cm3.

Given,

• Radius of the sphere is v√3.

• Volume of cylinder is maximum.

Let us consider,

• The radius of the sphere be 'R' units.

• Volume of the inscribed cylinder be 'V'.

• Height of the inscribed cylinder be 'h'.

• Radius of the cylinder is 'r'.

At present permit Air conditioning² = AB² + BC², hither Ac = 2R, AB = 2r, BC = h,

And so 4R^two = 4r^two + hⁱⁱ

r² = 1/4 [4R^two - h^two]----- (ane)

Let us consider, the volume of the cylinder:

Five = πr²h

Now substituting (ane) in the book formula,

For finding the maximum/ minimum of given function, nosotros tin can find information technology past differentiating it with h and and so equating it to cipher. This is because if the role Five(h) has a maximum/minimum at a point c then 5'(c) = 0.

Differentiating the equation (2) with respect to h:

To find the disquisitional point, we need to equate equation (3) to aught.

[as h cannot exist negative]

Now to check if this disquisitional point will determine the maximum volume of the inscribed cone, we need to check with 2d differential which needs to be negative.

Consider differentiating the equation (3) with h:

Now allow us find the value of

and then the function V is maximum at h = ten

Substituting h in equation (1)

Equally 5 is maximum, substituting h and r in the book formula:

V = π (l) (10)

V = 500π cmⁱⁱⁱ

Therefore when the volume of a inscribed cylinder is maximum and is equal 500π cm^three

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Source: https://www.sarthaks.com/1175841/show-that-the-maximum-volume-the-cylinder-which-can-be-inscribed-in-sphere-radius-53-500