Sorry for my poor English in advance. If there is any problem with my description, please tell me, thanks!
I suppose the thickness of this bucket as $a$(the question doesn't say it has, but I think a bucket should have thickness)
So the Surface Area $S$ will be $S = \pi(r+a)^2 + [\pi(R+a)^2-\pi R^2] + \left\{ \frac{1}{2}[\frac{R+a}{R-r} \sqrt{(R-r)^2 + (h + a)^2}]^{2} \frac{2 \pi (R - r)}{\sqrt{(R-r)^2 + (h + a)^2}} - \frac{1}{2}[\frac{r+a}{R-r} \sqrt{(R-r)^2 + (h + a)^2}]^{2} \frac{2 \pi (R - r)}{\sqrt{(R-r)^2 + (h + a)^2}} \right\} \\ = 2 \pi (R + r)a + \pi r^2 + 2 \pi a^2 + \pi \sqrt{(R-r)^2 + (h+a)^2}(R+r+2a) \\ = Const.$
Volume $V$ will be
$V = \frac{1}{3} \pi R^2(\frac{R}{R-r} h) - \frac{1}{3} \pi r^2(\frac{r}{R-r} h) \\ = \frac{1}{3} \pi h(R^2+Rr+r^2)$
Then I want to use Lagrange multiplier to find its maximum, but there are four unknown in $S$ and only three in $V$, so I don't know whether I can write the equation below or not.
$$\begin{cases} V_R = \lambda S_R \\ V_r = \lambda S_r \\ V_h = \lambda S_h \\ V_a = \lambda S_a \end{cases} $$
where $V_a = 0$.
Another question is I guess I can't get a precise answer? If that's right, how can I get an approximate answer? Using excel or anything else?
Hint. Ignoring the thickness, we see that the surface area is given by $$πr^2+π(R+r)\sqrt{h^2+(R-r)^2}=k,$$ for some positive constant $k.$ Also, volume is given by $$\frac13π(R^2+Rr+r^2).$$ Thus the volume may be expressed only in terms of the radii as $$\frac13π\frac{R^2+Rr+r^2}{R+r}\sqrt{(k-R^2)(k-R^2-2r^2)},$$ and the partial derivatives of this expression with respect to $R$ and $r$ may be relatively easily obtained via taking logarithms first, and then setting these to zero to solve for the possible extrema of the function and the values of the radii that extremize the volume.