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Modeling a sphere in a cone 
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#1
Jan211, 08:56 PM

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1. The problem statement, all variables and given/known data
This is an optimization problem but I'm having trouble modeling the question. There is a sphere encased in a cone. The sphere has a fixed radius R and the cone has a variable height h and radius r. There is also a variable angle theta at the base of the cone. Express the volume of the cone as a function of the angle theta, then find what slant angle theta should be used for the volume to be a minimum. 2. The attempt at a solution So far I have V=(pi*r^3*tan(θ))/3, where tan(θ)=h/r the problem is how to describe r in terms of θ thanks in advance 


#2
Jan211, 09:35 PM

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There is another right triangle in the problem. Drop a perpendicular from the center of the sphere to the cone. One leg is R. The other is sqrt(r^2+h^2). And that triangle splits into two similar triangles. Does that help? This is kind of similar to your other problem.



#3
Jan211, 09:48 PM

P: 196

I understand that there are two similar triangles in the cone but what I'm wondering is how to get the equation for volume solely in terms of theta.



#4
Jan211, 09:59 PM

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Modeling a sphere in a cone



#5
Jan211, 11:11 PM

P: 196

I feel like i've worked this one out every which way and I'm not seeing a relationship where i get something that is equivalent to the h*r^2 in terms of theta, unless that's where I'm going wrong?



#6
Jan311, 07:58 AM

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#7
Jan311, 04:55 PM

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Using the proportionality of similar triangles, I have:
R=(r(hr))/h R=(r*sqrt(h2R^2))/sqrt(r^2+h^2) So, if tan theta = h/r = (sqrt(h2R^2)/R), I'm having trouble seeing an identity for R that will give me the equivalent of r^2 or r*h^2 that will give me the r^2h I need to express the volume in terms of theta and R 


#8
Jan311, 05:43 PM

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