11.5. In the tetrahedron $S A B C$, the edges $S A, S B, S C$ are mutually perpendicular and equal to $a, b, c$ respectively. a) Find the side of the cube with vertex $S$ of the largest volume that is entirely contained within the tetrahedron; b) Determine the dimensions of the rectangular parallelepiped with vertex $S$ of the largest volume that is entirely contained within the tetrahedron.
Answer: a) $\frac{a b c}{a b+b c+a c} ;$ b) $\frac{a}{3}, \frac{b}{3}, \frac{c}{3}$. Solution. a) Analogous to the solution of problem 9.5, consider a rectangular coordinate system with the origin at point $S$ and coordinate axes along $S A, S B$, $S C$. Then the equation of the plane $A B C$ is $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ (this is the equation of the plane passing through three points $A(a, 0,0), B(0, b, 0), C(0,0, c))$. The diagonal of the cube passing through $S$ has the equation $x=y=z$. Substituting it into the equation of the plane $A B C$, we get the point of intersection: $x\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=1 \Leftrightarrow x=y=z=\frac{a b c}{a b+b c+a c}$.

b) Analogous to problem 9.5, we have a problem of maximizing the function of volume $V=x y z$ under the condition $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$. This maximum is obtained from the inequality of means for three numbers: $V=x y z=a b c \cdot\left(\frac{x}{a} \cdot \frac{y}{b} \cdot \frac{z}{c}\right) \leq a b c \cdot\left(\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right) / 3\right)^{3}=\frac{a b c}{27}, \quad$ it $\quad$ is $\quad$ achieved, $\quad$ when $\frac{x}{a}=\frac{y}{b}=\frac{z}{c}=\frac{1}{3}$.