2. A mobile railway platform has a horizontal bottom in the form of a rectangle 8 meters long and 5 meters wide, loaded with grain. The surface of the grain has an angle of no more than 45 degrees with the base plane (otherwise the grains will spill), the density of the grain is 1200 kg/m³. Find the maximum mass of grain loaded onto the platform.
Answer: 47.5 t. Solution. The calculation shows that the maximum height of the grain pile will be half the width of the platform, that is, 2.5 m. The pile can be divided into a "horizontally lying along the platform" prism (its height is 3 m, and the base is a right-angled isosceles triangle with legs $\frac{5 \sqrt{2}}{2}$ and hypotenuse 5), and two identical pyramids at the "ends" of the platform (the base of each is a rectangle 5 x 2.5 and the height is 2.5).

The total volume is $\frac{1}{2} \cdot 5 \cdot \frac{5}{2} \cdot 3 + 2 \cdot \frac{1}{3} \cdot 5 \cdot \frac{5}{2} \cdot \frac{5}{2} = \frac{75}{4} + \frac{125}{6} = \frac{25 \cdot 19}{12} \mathrm{m}^{3}$. Therefore, the mass is $\frac{25 \cdot 19}{12} \cdot 1200 = 47500$ kg.

Grading criteria: 20 points: correct solution and correct answer; 15 points: with minor defects in an otherwise correct solution and correct answer; 10 points: correct solution, but the answer is incorrect due to an arithmetic error.