16.10. (CPR, 58). Let $S$ and $V$ denote the surface area and volume of a regular $n$-sided pyramid.

a) For given values of $n$ and $S$, find the maximum value of $V$.

b) Find the side lengths of the bases and the heights of all pyramids for which $n=4, S=144, V=64$.
16.10. Let's introduce additional notations: $Q$ - the area of the base, $h$ - the height of the pyramid, $x$ - the cosine of the dihedral angle between the base and a lateral face, $a$ - the side of the base, $r$ - the radius of the circle inscribed in the base. Then the following equalities hold:

$$
\begin{aligned}
& a=2 r \operatorname{tg}\left(180^{\circ} / n\right), Q=n(1 / 2) r a=n r^{2} \operatorname{tg}\left(180^{\circ} / n\right), \\
& h=r \operatorname{tg}(\arccos x)=r \sqrt{1-x^{2} / x,} S=Q+(1 / x) Q
\end{aligned}
$$

from which

$$
\begin{aligned}
& Q=x S /(x+1), r=\sqrt{\frac{Q}{n \operatorname{tg}\left(180^{\circ} / n\right)}}=\sqrt{\frac{S}{n \operatorname{tg}\left(180^{\circ} / n\right)} \frac{x}{x+1}}, \\
& V=\frac{1}{3} h Q=\frac{1}{3} \frac{\sqrt{1-x^{2}}}{x} \cdot \sqrt{\frac{S}{n \operatorname{tg}\left(180^{\circ} / n\right)} \frac{x}{x+1}} \cdot \frac{x S}{x+1}= \\
& =\frac{1}{3} \frac{-S^{3 / 2}}{\sqrt{n \operatorname{tg}\left(180^{\circ} / n\right)}} \cdot f(x)
\end{aligned}
$$

where

$$
f(x)=\frac{\sqrt{x(1-x)}}{1+x}
$$

a) We have

$$
f^{\prime}(x)=\frac{1-3 x}{2(1+x)^{2} \sqrt{x(1-x)}},
$$

therefore the maximum value of the function $f(x)$ for $x \in(0 ; 1)$ is $f(1 / 3)=\sqrt{2} / 4$ (since $f^{\prime}(x)>0$ for $x<1 / 3$), and the desired value of the volume $V$ is

$$
\frac{\sqrt{2}}{12} \cdot \frac{S^{3 / 2}}{\sqrt{n \operatorname{tg}\left(180^{\circ} / n\right)}}
$$

b) Substituting the given values of $n, S, V$ into the obtained relationship between the quantities $V$, $S, n, x$, we get the equation for $x$

$$
\sqrt{x(1-x)} /(1+x)=2 / 9
$$

which has roots $x_{1}=1 / 17$ and $x_{2}=4 / 5$. Therefore, $Q_{1}=8$ and $Q_{2}=64$, from which $r_{1}=\sqrt{2}$ and $r_{2}=4$, hence,

$$
a_{1}=2 \sqrt{2}, h_{1}=24 \text { and } a_{2}=8, h_{2}=3
$$