find dimensions of largest cone that can be inscribed into sphere of radius r

[dlguzman06]

find dimensions of the largest cone that can be inscribed into a sphere with the radius of "r"?

therefore r is a constant. PLEASE show work or I wont be able to learn. If u want to work out on paper and add pic of your work that would be amazing, thank you so much for any help.

I realize "r" is a constant and draw out my picture and i know i need to come up with a volume function for the cone with respect to the sphere's radius, therefore having only one variable. I also know I will have to take the derivative of the volume function and get the critical value so that i can find the maximum. So hence i know some steps to take i just dont know how to take them. please help. :-|

 

[Deleted member 4993]

find dimensions of largest cone that can be inscribed into sphere of radius r

I realize "r" is a constant and draw out my picture and i know i need to come up with a volume function for the cone with respect to the sphere's radius, therefore having only one variable. I also know I will have to take the derivative of the volume function and get the critical value so that i can find the maximum. So hence i know some steps to take i just don't know how to take them. just a boost would be appreciated please. I will take hints, anything. :?

Start with the problem of finding largest triangle (area wise) inside a given circle. This problem will be somewhat similar - not exactly.

 

[MarkFL]

I would orient the \(\displaystyle x\)-axis such that it coincides with the axis of symmetry of the cone, and put the apex of the cone at the origin. Now, letting \(\displaystyle h\) be the distance along this axis from the apex to the base of the cone, and \(\displaystyle r_C\) be the radius of the base, we may state:

\(\displaystyle (h-r)^2+r_C^2=r^2\)

Now, simplify this, solve for \(\displaystyle r_C^2\) and substitute for this into the volume of the cone:

\(\displaystyle V_C=\frac{1}{3}\pi r_C^2h\)

and you will have a function for the volume in one variable, \(\displaystyle h\). Then apply your optimization method.

 

[HallsofIvy]

find dimensions of the largest cone that can be inscribed into a sphere with the radius of "r"?

therefore r is a constant. PLEASE show work or I wont be able to learn.

That's a really sad thing to say. Do you realize that you are saying that you can only memorize what other people do and are not capable of thinking for yourself? I don't believe that true but it is sad that you say it.