5. At the base of the quadrilateral pyramid $S M N K L$ lies a rhombus $M N K L$ with a side length of 4 and an acute angle $N M K$ of $60^{\circ}$. It is known that $S M=2, S N=4$. Determine the values of the lengths of the remaining two edges $S K$ and $S L$ for which the volume of the pyramid reaches its maximum, and calculate this volume.
Answer: $V_{\max }=4 \sqrt{5}, S K=3 \sqrt{2}, S L=\sqrt{46}$.

## 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.

Therefore, $S P$ and $S H$ must coincide, and consequently,

![](https://cdn.mathpix.com/cropped/2024_05_06_498048f983b1f7dad400g-5.jpg?height=460&width=471&top_left_y=1966&top_left_x=1521)
the faces $S M N$ and $M N K L$ are perpendicular.

Triangle $S M N$ is isosceles because $S N=4, M N=4$.

After elementary calculations, we get $S P=\frac{\sqrt{15}}{2}, M P=\frac{1}{2}, N P=\frac{7}{2}$.

From triangle $M P K$, we find $P K^{2}=M K^{2}+M P^{2}-2 M K \cdot M P \cdot \cos 60^{\circ} \Rightarrow \quad P K^{2}=\frac{57}{4}$.

From the right triangle $S P K$, we find $S K^{2}=S P^{2}+P K^{2} \Rightarrow S K^{2}=18 \Rightarrow$ $S K=3 \sqrt{2}$.

From triangle $P N L$, we find $P L^{2}=N L^{2}+P N^{2}-2 N L \cdot P L \cdot \cos 120^{\circ} \Rightarrow P L^{2}=\frac{169}{4}$.

From the right triangle $S P L$, we find $S L^{2}=S P^{2}+P L^{2} \Rightarrow S L^{2}=46 \Rightarrow$ $S K=\sqrt{46}$.

$$
V_{\max }=\frac{1}{3} \cdot S_{M N K L} \cdot S P=\frac{1}{3} \cdot 4 \cdot 4 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{15}}{2}=4 \sqrt{5}
$$

## Grading criteria are provided in the table:

| Points | Grading 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, diagrams, 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. |