Expert: Alon Mandes - 2/2/2009

Question

Thanks alot for your response, but it seems, I still had problems with the notation, so I am writing this follow up in a new message, as Ask Experts tells me I have too many follow ups.

Anyway, you wrote:

"Answer   From the graph of A(h) we can easily see that h=5.25 ("which is :
2R*(5-√5)/√10) I took R=5 in this case") is a max. The graph of
A'(h) is zero in this particulary h.

Alon."

Actually I don't quite see how you get 5.25 from the above, but that may not be relevant as it appears that I still have some problems with notation

The stationary points are

h=2R*√(5-√5)/√10 <-- (This is NOT 2R*(5-√5)/√10 as I previously wrote. It looks like I left out one of the square roots)

and
r=R√(5+√5)/√10

and

h=2R*√(5+√5)/√10

r= R√(5-√5)/√10

Now from plugging in the solutions into the original equation (and yes I guess from the graph) it seems obvious that h=2R*√(5-√5)/√10 is indeed the maximum.

The problem is that

h=2R*√(5+√5)/√10 also gives me a negative in my second derivative.

I would have expected that the second derivative would be positive, or possibly not definable, but I have checked my arithmetic numerous times, and keep coming up with the same answer. I have also gone over the derivation of the second derivative many times and cannot figure out where I have gone wrong.

For reference here is the second derivative I got.

A"= pi*(  -1 + (-12R^2h+2h^3)/(4R^2-h^2)(√(4R^2-h^2) )

And here is the link to the original thread.

http://en.allexperts.com/q/Calculus-2063/2009/1/Applied-Maximum-Problem-Surface....

I hope I am not abusing this forum by continuing to be so persistent in trying to understand this problem. You have indeed been very helpful.

Regards,

Richard @btw, did you try to send me an email message directly? If so I never got it. I only ask because my provider has an aggressive stance towards spam, the result being that sometimes non-spam messages get rejected.  

Answer
The solution 5.25 is gained when we set R=5. To present a numerical graph we need to insert numbers..So consider it as an example : R=5.
The 2nd derivative is indeed negative for all h. This can be seen in
the graph. Our stationary point is 2R*√(5-√5)/√10. As for the 2nd
solution (the possitive one, I'll chick it again).
Regards,
Rami.