A hemisphere of radius 7 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.

The radius and height of the cylinder of maximum volume are approximately 4.041 and 5.715, respectively.

The volume of the hemisphere () is described by the following expression:

(1)

Where:

The volume of the cylinder inscribed in the hemisphere () is:

(2)

Where:

By Pythagorean theorem we derive a relationship between the radius of the hemisphere and the radius of the cylinder:

(3)

By applying (3) in (1) and simplifying the resulting expression we have this outcome:

(4)

Now we perform first and second derivative tests to determine the dimensions of the cylinder so that volume found is a maximum:

FDT

(5)

SDT

Since , indicates that critical value found in the previous step is associated to a maximum.

By (3) we have an expression for the radius of the cylinder:

(6)

If we know that , then the radius and the height of the cylinder of maximum volume are:

,

The radius and height of the cylinder of maximum volume are approximately 4.041 and 5.715, respectively.

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radius r = 5.72

Height h = 4.04

Step-by-step explanation: Please find the attached file for the solution

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step-by-step explanation: