4. Three pulleys with parallel axes and identical radii $r=2$ cm must be connected by a flat belt drive. The distance between the axes of rotation of pulleys $O_{1}$ and $O_{2}$ is 12 cm, and the distance between the axes of rotation of pulleys $O_{1}$ and $O_{3}$ is 10 cm. The distance from the axis $O_{3}$ to the plane in which the axes $O_{1}$ and $O_{2}$ are located is 8 cm. Determine the length of the belt for the drive, which is made by sewing the ends of an inextensible rubberized cord (assume that the length of the belt is equal to the length of this cord). Will a cord 54 cm long always be sufficient for its manufacture?
Solution. 1) First, we solve the geometric problem of finding the side $\mathrm{O}_{2} \mathrm{O}_{3}$. According to the condition, $\sin \angle O_{1}=\frac{8}{10}=\frac{4}{5}$; therefore, $\cos \angle O_{1}= \pm \frac{3}{5}$. The third side is found using the cosine theorem and equals either 10 cm or $2 \sqrt{97}$ cm.

2) The perimeter of the figure will be the sum of three straight segments and the sum of three arcs. The sum of the straight segments is equal to the perimeter of the triangle $O_{1} O_{2} O_{3}$.
3) The sum of the lengths of the three arcs is $r \cdot(\pi-\alpha)+r \cdot(\pi-\beta)+r \cdot(\pi-\gamma)$ (where $\alpha, \beta, \gamma$ are the angles of the triangle $\left.O_{1} O_{2} O_{3}\right)$, which equals $r \cdot(3 \pi-\alpha-\beta-\gamma)=r \cdot(3 \pi-\pi)=2 \pi r$.
4) In the case of an acute triangle, the required length of the belt is $12+10+10+2 \pi \cdot 2=32+4 \pi$ cm (for reference: $32+4 \pi \approx 44.57$). This is obviously less than 54, so the string is sufficient.
5) In the case of an obtuse triangle, the required length of the belt is $12+10+2 \sqrt{97}+2 \pi \cdot 2=22+2 \sqrt{97}+4 \pi$ cm (for reference: $22+2 \sqrt{97}+4 \pi \approx 54.26$). Here, the string is not sufficient.

Therefore, the answer is: no. To get the maximum score, a careful comparison is required: $22+2 \sqrt{97}+4 \pi>54 \Leftrightarrow \sqrt{97}+2 \pi>16 \Leftrightarrow \sqrt{97}+2 \pi>9.8+6.2=16$.

Answer: a) $32+4 \pi$ cm or $22+2 \sqrt{97}+4 \pi$ cm; b) not always.

Grading criteria: 20 points - correct (not necessarily the same as above) solution and correct answer; 15 points - mostly correct solution and correct answer, but there are defects (for example: no comparison or incorrect comparison of numbers in the last point of the solution; it is not proven that the sum of the lengths of the arcs is equal to the circumference); 10 points - the solution is mostly correct, but contains computational errors; 5 points - only one of the two cases is considered, and the solution and answer (for this case) are completely correct.