This one is kinda' related. Read through it and see if you can construct a route to your problem:
http://www.mathhelpforum.com/math-he...al-volume.html
Question: A right cylinder is inscribed in a sphere with a radius of 5. Determine the maximum volume of the cylinder.
Answer: Volume ≈ 302
I'm having problems visualizing and even drawing this thing in the first place. The diameter of the sphere is not equal to the height cylinder, right? Apparently, I need to construct a right angle triangle in there but what do I do then?
Any help would be GREATLY appreciated!
Thanks in advance!
This one is kinda' related. Read through it and see if you can construct a route to your problem:
http://www.mathhelpforum.com/math-he...al-volume.html
1. See attachment.
2. With your question R = 5.
3. The volume of a cylinder is
4. You'll get the volume of the cylinder as a function wrt h:
5. Differentiate V(h) wrt h and solve the equation V'(h) = 0 for h. I've got (Of course there is a negative solution too, but a negative radius of a sphere isn't very plausible)
6. Plug in this value into the equation of the function:
7. The exact value is
Now, I get how to do it and I get the final answer! But...what is the restriction on r (cylinder's radius) to prove that it is indeed the maximum. Would the 1st and second derivative tests work as well? Or would those just tell me that it is a local maximum or minimum and not if it is the absolute maximum or minimum?
1. The 2nd derivative is negative for all positive values of h. Thus you have found a maximum.
2. The radius of the cylinder varies between 0 and R: If r = 0 then the volume of the cylinder is 0, which is quite a minimal value for a volume. If r = R then the volume of the cylinder is 0 too, which is another minimal value for a volume. Since the V-function is continuous and differentiable you know that between 2 minimums (is that the correct expression? It sounds like a very small mother) is a maximum (big mother?)