Problem 3. Experimenters Glafira and Gavriil placed a triangle made of thin wire with sides of 30 mm, 40 mm, and 50 mm on a white flat surface. This wire is covered with millions of mysterious microorganisms. The scientists found that when an electric current is applied to the wire, these microorganisms begin to move chaotically on this surface in different directions at an approximate speed of $\frac{1}{6}$ mm/sec. As they move, the surface along their path is stained red. Find the area of the stained surface after 1 minute of current application. Round it to the nearest whole number of square millimeters.
Solution. In one minute, the microorganism moves 10 mm. Since in a right triangle with sides $30, 40, 50$, the radius of the inscribed circle is 10, all points inside the triangle are no more than 10 mm away from the sides of the triangle. Therefore, the microorganisms will fill the entire interior of the triangle.

When moving outward, points 10 mm away from the sides of the triangle and points 10 mm away from the vertices will be reached.

In the end, the total occupied area is: the area of the triangle + three strips 10 mm wide each, located outside the triangle, with a total length equal to the perimeter of the triangle + 3 circular sectors with a radius of 10, which together form a circle. We get:

$$
\frac{30 \cdot 40}{2} + 10 \cdot (30 + 40 + 50) + \pi \cdot 10^{2} = 600 + 1200 + 100 \pi = 1800 + 100 \pi \approx 2114 \mathrm{Mm}^{2}
$$

Answer: $1800 + 100 \pi \approx 2114$ mm $^{2}$.