5. The base of the quadrilateral pyramid $S M N K L$ is a rectangle $M N K L$. The lengths of four edges of the pyramid are known: $M N=5, N K=2, S M=3, S N=4$. Determine the values of the lengths of the remaining two edges $S K$ and $S L$ for which the volume of the pyramid is maximized, and calculate this volume.
Answer: $V_{\max }=8, S K=2 \sqrt{5}, S L=\sqrt{13}$.

## Solution.

Drop perpendiculars $S P$ from point $S$ to edge $M N$ and $S H$ to the plane of the base $M N K L$.

All sides of triangle $S M N$ are known, so the length of $S P$ is uniquely determined. The base of the pyramid is fixed, so the volume reaches its maximum value if the height $S H$ of the pyramid is maximized.

This means that $S P$ and $S H$ must coincide, and therefore,

![](https://cdn.mathpix.com/cropped/2024_05_06_498048f983b1f7dad400g-2.jpg?height=463&width=528&top_left_y=1936&top_left_x=1461)
the faces $S M N$ and $M N K L$ are perpendicular.

Triangle $S M N$ is a right triangle because $S M=3, S N=4, M N=5$.

After elementary calculations, we get $S P=\frac{12}{5}, M P=\frac{9}{5}, N P=\frac{16}{5}$.

From the right triangles $N K P$ and $S K P$, $M L P$ and $S L P$, we find

$$
\begin{gathered}
S K^{2}=S P^{2}+N P^{2}+N K^{2}=20, S L^{2}=S P^{2}+M P^{2}+M L^{2}=13 \Rightarrow S K=2 \sqrt{5}, S L=\sqrt{13} \\
V_{\max }=\frac{1}{3} \cdot S_{M N K L} \cdot S P=\frac{1}{3} \cdot 10 \cdot \frac{12}{5}=8 .
\end{gathered}
$$

The evaluation criteria are provided in the table:

| Points | Evaluation Criteria |
| :---: | :--- |
| $\mathbf{7}$ | Complete and justified solution. |
| $\mathbf{6}$ | Justified solution with minor flaws. |
| $\mathbf{5 - 6}$ | The solution contains minor errors, gaps in justification, but is generally correct and can become fully correct after minor corrections or additions. |
| $\mathbf{4}$ | The problem is more solved than not, for example, one of the two (more complex) significant cases is correctly considered. |
| $\mathbf{2 - 3}$ | The problem is not solved, but formulas, drawings, considerations, or auxiliary statements relevant to the solution of the problem are provided. |
| $\mathbf{1}$ | The problem is not solved, but an attempt at a solution has been made, for example, individual (partial) cases have been considered in the absence of a solution or in the case of an incorrect solution. |
| $\mathbf{0}$ | No solution, or the solution does not meet any of the criteria listed above. |

## Ministry of Science and Higher Education of the Russian Federation  Council of Rectors of Universities of Tomsk Region  Open Regional Interuniversity Olympiad 2021-2022  MATHEMATICS (11th grade)  Final Stage  Variant 2