SOLUTION: Find the dimensions of the right circular cylinder of greatest volume that can be inscribed in a right-circular cone with a radius of 5 cm and a height of 12 cm.

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Question 1191566: Find the dimensions of the right circular cylinder of greatest volume that can be inscribed in a right-circular cone with a radius of 5 cm and a height of 12 cm.
Answer by greenestamps(13200) About Me  (Show Source):
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Here is a 2-dimensional sketch of a right circular cylinder inscribed in a right circular cone.



We are given AD = 5 and CD = 13, so we can conclude AC = 12.

Let x be the radius of the cylinder, BE. Similar triangles ACD and BCE give us BC = (12/5)x = 2.4x. The height of the cylinder is then 12-2.4x.

Express the volume of the cylinder in terms of x and maximize the volume by finding where the derivative of the volume function is 0.

V=%28pi%29%28r%5E2%29%28h%29=%28pi%29%28x%5E2%29%2812-2.4x%29=pi%2812x%5E2-2.4x%5E3%29

dV%2Fdx+=+pi%2824x-7.2x%5E2%29+=+%28pi%29%2824x%281-.3x%29%29

.3x+=+1
x+=+1%2F.3+=+10%2F3

The volume is maximum when the radius of the cylinder is 10/3.

That makes the height 12-2.4%2810%2F3%29=12-8=4

ANSWER: radius 10/3; height 4