Aright circular cylinder is inscribed in a sphere of radius r. find the dimensions of such a cylinder with the largest possible volume (your answer may depend on r).

Largest volume in terms of r =  4r^3π/ 3√3 or you could write it as  4√3r^3π/9.

Step-by-step explanation:

The height and radius of the right circular cylinder inscribed in sphere are and , respectively.

Geometrically speaking, the volumes of the sphere () and the right circular cylinder () are defined by the following formulae:

Sphere

(1)

Cylinder

(2)

Where:

In addition, we find the following relationship by the Pythagorean theorem:

(3)

By (3) we clear :

(3) in (2):

(4)

By (1):

(1) in (4):

(5)

Now we proceed to perform first and second derivative tests:

FDT

(6)

SDT

(7)

Since and , the critical value is associated to the maximum volume of the cylinder.

The dimensions of the right circular cylinder are the following:

By (6):

By (3):

The height and radius of the right circular cylinder inscribed in sphere are and , respectively.

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step-by-step explanation: