I'm trying to solve an equation, and while it work, "plain" Solve returns two solutions that are, in the specific case, of no interest.

Clear[r, h, V]
V[r_, h_] := \[Pi] r^2 h
A[r_, h_] := \[Pi] r^2 + 2 \[Pi] r h
h0 = h /. Solve[V[r, h] == V0, h][[1]]
D[A[r, h0], r]
Solve[D[A[r, h0], r] == 0, r]

So I tried, as suggested on some web pages,

Solve[D[A[r, h0], r] == 0 && r > 0, r]

but Mathematica returns {}.

What is the correct syntax to get the correct solution? TIA.

(N.B.: Though probably not important for the question, the mathematical problem above is about how to find the radius $r$ and height $h$ for a circular cylinder with a bottom (no top) that consumes the least material given a volume $V_0$.)


Add the information that v0 > 0:

Solve[D[A[r, h0], r] == 0 && r > 0 && V0 > 0, r][[1]]

{r -> ConditionalExpression[Root[-V0 + π #1^3 &, 1], V0 > 0]}

Or specify the domain as PositiveReals:

Solve[D[A[r, h0], r] == 0, r, PositiveReals][[1]]

{r -> ConditionalExpression[Root[-V0 + π #1^3 &, 1], V0 > 0]}

 FullSimplify[%, V0 > 0]

{r -> V0^(1/3)/π^(1/3)}

  • $\begingroup$ Thanks! To get the $r$ assigned to a variable $r_0$ I wrote r0 = r /. FullSimplify[%, V0 > 0][[1]]. Is there some 'shorter' syntax to 'extract' the only value when the (double) output list is one item only so I don't have to type [[1]]? Trying to understand the FullSimply; why do I have to specify V0>0 again since it was given in the Solve command and also in the ConditionalExpression command? $\endgroup$
    – mf67
    Aug 4 '19 at 21:57
  • $\begingroup$ @mf67, in don't know of any way to avoid using [[1]] (you can put after Solve or after FullSimplify). ConditionalExpresssion does not eliminate the condition V0>0; so wee need to use FullSimplify using the same condition again. $\endgroup$
    – kglr
    Aug 4 '19 at 22:40

From a mathematical point of view this type of problem is sometimes more easily solved using Lagrange multipliers for the constraints,

V[r_, h_] := \[Pi] r^2 h
A[r_, h_] := \[Pi] r^2 + 2 \[Pi] r h + \[Lambda] (V[r, h] - V0)

solns = Solve[{D[A[r, h], r] == 0, D[A[r, h], h] == 0, D[A[r, h], \[Lambda]] == 0}, {r, h, \[Lambda]}] 

soln = Map[Reduce[Join[#, {r > 0, h > 0, V0 > 0}]] &, solns /. Rule -> Equal]

Cases[soln, Except[False]]

(* {V0 > 0 && \[Lambda] == (-2*Pi^(1/3))/V0^(1/3) && r == V0^(1/3)/Pi^(1/3) && h == V0^(1/3)/Pi^(1/3)} *)
  • $\begingroup$ That is interesting. The 4th line is however, for me, difficult to understand and I have not (yet) found a good tutorial for all these 'cryptic' commands that are very powerful, once you know them. $\endgroup$
    – mf67
    Aug 4 '19 at 23:28
  • $\begingroup$ To understand line 4, just pull it apart the same way it got put together: Evaluate solns /. Rule -> Equal, see what it does. Take out the Reduce[], evaluate and see what that does. If you don't know Map, see what Map[f, solns/.Rule->Equal] does. Finally, look up the documentation for Function (which will explain # and &). $\endgroup$ Aug 5 '19 at 0:01

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