Optimizing a Conical Frustum Using Partial Differentiation I am working on minimizing the surface area of a frustum with a known volume, however, the equations for the volume and surface area (listed below) have three variables. I am trying to use partial differentiation to optimize surface area with volume as a constraint, yet, differentiating with respect to any variable results in very messy work that can't be solved for a variable. Is there a better way to do it, or can anyone please show me how to do it using partial differentiation?
$
V=1/3(πh)(r^2+rR+R^2)
$
$
S.A=π(r+R)(\sqrt{(r-R)^2+h^2}+πr^2+πR^2
$
Thank you.
 A: Define the Lagrangian function
$$\Lambda(r,R,h;\lambda):=\pi(r+R) \sqrt{(r-R)^2+h^2}+\pi r^2+\pi R^2 -\lambda(V_0-(\pi h) (r^2+rR+R^2)/3), $$
with $V_0$ denoting the allocated volume.
The partial derivatives are
$$\begin{align}\Lambda_r&=\pi  \sqrt{h^2+(r-R)^2}+\frac{\pi  (r-R) (r+R)}{\sqrt{h^2+(r-R)^2}}+\frac{1}{3} \pi  h \lambda  (2 r+R)+2 \pi  r, \\ \Lambda_R&=\frac{\pi  \left(R^2-r^2\right)}{\sqrt{h^2+(r-R)^2}}+\pi  \sqrt{h^2+(r-R)^2}+\frac{1}{3} \pi  h \lambda  (r+2 R)+2 \pi  R, \\ \Lambda_h &=\frac{\pi  h (r+R)}{\sqrt{h^2+(r-R)^2}}+\frac{1}{3} \pi  \lambda  \left(r^2+r R+R^2\right), \\ \Lambda_\lambda&=\frac{1}{3} \pi  h \left(r^2+r R+R^2\right)-V_0. \end{align}$$
Proceed with a change of variables, motivated by the square root in the above
$$
\begin{align}
h &= \rho \cos \theta, \\
r &= \frac{s+\rho \sin \theta}{2}, \\
R &= \frac{s-\rho \sin \theta}{2}.
\end{align} 
$$
Setting the partial derivates above equal to zero, and applying the substitution gives
$$
\begin{align}
0&=\frac{1}{6} \pi  \lambda  \rho  \cos (\theta ) (\rho  \sin (\theta )+3 s)+\pi  (\sin (\theta )+1) (\rho +s),\\
0&=-\frac{1}{6} \pi  \lambda  \rho  \cos (\theta ) (\rho  \sin (\theta )-3 s)-\pi  (\sin (\theta )-1) (\rho +s), \\
0&=\frac{1}{24} \pi  \lambda  \left(-\rho ^2 \cos (2 \theta )+\rho ^2+6 s^2\right)+\pi  s \cos (\theta ),\\
0 &=\frac{1}{12} \pi  \rho  \cos (\theta ) \left(\rho ^2 \sin ^2(\theta )+3 s^2\right)-V_0.
\end{align}
$$
The first three equations can be written in linear form as
$$ 
\left(
\begin{array}{ccc}
 \pi  (\sin (\theta )+1) & \pi  (\sin (\theta )+1) & \frac{1}{6} \pi  \rho  \cos (\theta ) (\rho  \sin (\theta )+3 s) \\
 \pi  (1-\sin (\theta )) & \pi  (1-\sin (\theta )) & \frac{1}{6} \pi  \rho  \cos (\theta ) (3 s-\rho  \sin (\theta )) \\
 0 & \pi  \cos (\theta ) & \frac{1}{24} \pi  \left(-\rho ^2 \cos (2 \theta )+\rho ^2+6 s^2\right) \\
\end{array} 
\right) \begin{pmatrix} \rho \\ s \\ \lambda \end{pmatrix} = \begin{pmatrix} 0\\ 0 \\ 0 \end{pmatrix}.
$$
If the matrix in the above is invertible, we must have $\rho=0$, which implies a planar frustum ($h=0$). Hence, we can assume that the matrix is singular, meaning its determinant
$$\frac{1}{3} \pi ^3 \rho  \sin (\theta ) \cos ^2(\theta ) (\rho -3 s)=0. $$
This vanishing determinant condition reduces the search to two seemingly admissible cases:

*

*$\rho=3s \implies h =2 \sqrt{(2 r+R) (r+2 R)}$

*$\rho \sin \theta =0 \implies r=R$

Case 1:
The first three equations, upon divisions allowed by $s>0, \cos \theta \neq 0$ become
$$\frac{1}{2} \pi  (3 \lambda  s \cos (\theta )+8) =0, \\\frac{1}{2} \pi  (3 \lambda  s \cos (\theta )+8)=0, \\\pi  \cos (\theta )+\frac{1}{8} \pi  \lambda  s (5-3 \cos (2 \theta ))=0. $$
It takes a little work to deduce from this that $\sin^2 \theta = \frac{1}{9}$, which gives $r=0$ or $R=0$. This renders Case 1 inadmissible.

Case 2:
Let us return to the Cartesian form of the system, assuming that $r=R$:
$$\pi  (h \lambda  r+h+2 r)=0, \\ \pi  (h \lambda  r+h+2 r)=0, \\\pi  r (\lambda  r+2)=0, \\\pi  h r^2-V_0 .$$
Since $r>0$ the third equation immediately gives $\lambda = -2/r$. It follows from there that $h=2r$ and finally that $r=\sqrt[3]{\frac{V_0}{2 \pi }}.$

Since there is a unique critical point, it has to be the minimum. That is, the frustum of volume $V_0$, of least surface area is a cylinder with radii $R=r=\sqrt[3]{\frac{V_0}{2 \pi }}$ and height $h=2\sqrt[3]{\frac{V_0}{2 \pi }}$.
Remark: This is somewhat expected based on the iso-perimetric inequality.
