Optimizing Volume of a Cone Encasing a Sphere: Finding the Minimum Slant Angle

[armolinasf]

Homework Statement

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

 

[Dick]

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.

 

[armolinasf]

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.

 

[Dick]

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.

You'll have to work it out. But if you know the angle is theta and the opposite leg is R, then you know everything about that triangle. Now look at the smaller triangle. The hypotenuse is R and an angle is theta. You know everything about that one as well, like r. You can express everything in terms of R and theta, right?

 

[armolinasf]

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?

 

[Dick]

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?

You aren't going to find an expression purely in terms of theta. You'll need to use the fixed radius R as well. You've already got one relationship. r/h=tan(theta). That let's you get rid of one variable. You just need one more. What have you found?

 

[armolinasf]

Using the proportionality of similar triangles, I have:

R=(r(h-r))/h

R=(r*sqrt(h-2R^2))/sqrt(r^2+h^2)

So, if tan theta = h/r = (sqrt(h-2R^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

 

[Dick]

Using the proportionality of similar triangles, I have:

R=(r(h-r))/h

R=(r*sqrt(h-2R^2))/sqrt(r^2+h^2)

So, if tan theta = h/r = (sqrt(h-2R^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

Look at the triangle whose vertices are i) the apex of the cone, ii) the center of the sphere and iii) a perpendicular from the center of the sphere to the side of the cone. I think that will give you an easier relation to work with.