[Guest]

A cone of revolution has an inscribed sphere tangent to the base of the cone (and to the sloping surface of the cone). A cylinder is circumscribed about the sphere so that its base lies in the base of cone.

The volume of the cone is V₁ and the volume of the cylinder is V₂.

Question: What is the minimum possible value of V₁ / V₂?

PS: I do not know the answer but I have calculated a value. If you post your answer for part 2, pls also show how you arrived at this value.

Edited August 5, 2009 by DeeGee

[Guest]

PS: I do not know the answer but I have calculated a value. If you post your answer for part 2, pls also show how you arrived at this value.

There is of course only 1 part. I had earlier posted two questions but the first one seemed rather simple.

Edited August 5, 2009 by DeeGee

[Guest]

The solution is 4/3. If the sphere has radius a, the volume of the cylinder is fixed at V = Pi*a^2*h = 2*Pi*a^3. The problem reduces to finding the minimum volume for a cone circumscribing a sphere. That is a well-known, but messy, first semester calculus problem. One solution can be found at

http://www.mathalino.com/reviewer/differential-calculus/64-65-maxima-and-minima-cone-inscribed-in-a-sphere-and-sphere-inscrib

The solution is that h = 4a and r^2 = 2*a^2, where r and h are the radius and height of the cone. The volume of the cone is then 1/3*Pi*r^2*h = 8/3*Pi*a^3. The ratio, then, is 4/3.

[HoustonHokie]

I think the best thing to do is first simplify to 2D to work out some relationships between the cone and sphere, and then work out the minimum ratio you seek.

Start with the picture below:

All the variable lengths can be described in terms of θ and R. So,

h = R/sin(θ)

H = R + h = R [1 + 1/sin(θ)]

B = 2H tan(θ) = 2R [1 + 1/sin(θ)] tan(θ)

Now, we can go back to 3D and plug these values into the formula for the volumes of a sphere and cone:

V_s = 4/3 πR³

V_c = H π (B/2)² / 3 = R³ [1 + 1/sin(θ)]³ [tan(θ)]² π / 3

So now everything is in terms of R and θ. R is a constant, so it doesn't really factor in, and it also makes the volume of the sphere a constant. And then, what we're really looking for is the value of θ that minimizes Vc. Well, you can take dVc/dθ, set it equal to 0, and solve for θ. Or you can do what I did and plug it into a spreadsheet. Either way, you get that θ = 19.47˚.

Taking the ratio of V_c/V_s, the R³ terms cancel out, and if θ = 19.47˚, then V_c/V_s = 2, which is what we're looking for.

[Guest]

I agree with mathman. I enclose how I arrived at an identical solution.

Incidentally, it is so much easier to make up a Word document using MathType than to use the primitive tools available on this site.

DeeGee Problem.doc