Problem 2. 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 at a distance from the sides of the triangle that does not exceed 10 mm. Therefore, the microorganisms will fill the entire interior of the triangle.

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

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

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

Answer: $1800 + 100 \pi \approx 2114$ mm².