2. A mobile railway platform has a horizontal bottom in the form of a rectangle 8 meters long and 4 meters wide, loaded with sand. The surface of the sand has an angle of no more than 45 degrees with the base plane (otherwise the sand grains will spill), the density of the sand is 1500 kg/m³. Find the maximum mass of sand loaded onto the platform.
Answer: 40 t. Solution. The calculation shows that the maximum height of the sand pile will be equal to half the width of the platform, that is, 2 m. The pile can be divided into a "horizontally lying along the platform" prism (its height is 4 m, and the base is an isosceles right triangle with legs $2 \sqrt{2}$ and hypotenuse 4), and two identical pyramids at the "ends" of the platform (the base of each is a rectangle 4 x 2 and the height is 2).

The total volume is $\frac{1}{2} \cdot 4 \cdot 2 \cdot 4 + 2 \cdot \frac{1}{3} \cdot 4 \cdot 2 \cdot 2 = 16 + \frac{32}{3} = \frac{80}{3}$ m $^{3}$. Therefore, the mass is $\frac{80}{3} \cdot 1500 = 40000$ 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.