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SquirrelHunte
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Homework Statement
Given a right circular cone you put an upside down cone inside it so that its vertex is at the center of the base of the larger cone, and its base is parallel to the base of the larger cone. If you choose the upside-down cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy?
Let H and R be the height and radius of the larger cone, let h and r be the height and radius of the small cone.
Homework Equations
Volume of a cone= V=(1/3)pi r^2 h
The Attempt at a Solution
Obviously I want the end problem to look like this:
Maximum (largest) volume of the small/Volume of the larger cone
So first I wanted to use the right angles to get similar triangles relating h and r. So I got h/H=r/R.
thus h=Hr/R or (H/R)r
then I put that into the volume formula of the smaller cone: (1/3)pi r^2 ((H/R)r)..
you end up with (H/3R)(pi)(r^3)=V
First derivative: (3H/3R)(pi)(r^2)...then...(H/R)(pi)(r^2)=V'
To get my maximum I set my V' to zero so (H/R)(pi)(r^2)=0 but my problem is that the only way to get that equation to equal zero is if r=0, but it can't equal 0 so now I am stuck...
Help Please?