A hemisphere of radius 8 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.

Answers

Answer 1

The dimensions of the cylinder that has the maximum volume can be

found be finding the maximum value of the volume function.

[tex]\mathrm{The \ \underline{height \ of \ the \ cylinder \ of \ maximum \ volume \ is \ \dfrac{8}{\sqrt{3} }}}[/tex][tex]\mathrm{The \ \underline{radius \ of \ the \ cylinder \ of \ maximum \ volume \ is \ \dfrac{8 \cdot \sqrt{6} }{3}}}[/tex]

Reasons:

The known parameter of the hemisphere are;

The radius of the hemisphere = 8

Required;

The radius and height of the maximum volume of the inscribed cylinder.

Solution:

Let h represent the height of the cylinder, and let r represent the radius of

the cylinder, we have;

r² + h² = 8²

Therefore;

r² = 8² - h² = 64 - h²

= 64 - h²

The volume of the cylinder, V = π·r²·h = π·(64 - h²)·h

V(h) = π·(64·h - h³)

At the maximum volume, we have;

[tex]\dfrac{dV(h)}{dh} = \dfrac{d}{dh} \left(\pi \cdot \left(64 \cdot h - h^3 \right) \right) = \pi \cdot \left(64 - 3 \cdot h^2 \right) = 0[/tex]

64 - 3·h² = 0

64 = 3·h²

[tex]h =\sqrt{\dfrac{64}{3} } = \dfrac{8}{\sqrt{3} }[/tex]

[tex]\underline{\mathrm{The \ height \ of \ the \ cylinder, \ h} = \dfrac{8}{\sqrt{3} }}[/tex]

[tex]r^2 = 64 - \dfrac{64}{3} = \dfrac{128}{3}[/tex]

[tex]r =\sqrt{\dfrac{128}{3}} = \dfrac{8 \cdot \sqrt{6} }{3}[/tex]

[tex]\underline{\mathrm{The \ radius \ of \ the \ cylinder, \ } r = \dfrac{8 \cdot \sqrt{6} }{3}}[/tex]

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Answer:

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