Task 5. (20 points) A two-meter gas pipe has rusted in two places. Determine the probability that all three resulting sections can be used as offsets for gas stoves, if according to regulations, the stove should not be located closer than 50 cm to the main gas pipe.

#
# Solution.

Let the sizes of the parts into which the pipe was cut be denoted as $x, y$, and $(200-x-y)$.

Obviously, the values of $x$ and $y$ can take any values from the interval $(0 ; 200)$. Then, the set of all possible combinations $(x, y)$ can be represented on the coordinate plane $O X Y$ as a right-angled triangle with sides equal to 200 cm (Fig. 2).

![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-32.jpg?height=565&width=826&top_left_y=417&top_left_x=615)

The measure of this set can be considered as the area of this triangle

$$
S=\frac{1}{2} \cdot 200 \cdot 200=20000 \mathrm{~cm}^{2}
$$

For the resulting parts to be suitable as flue pipes, the size of each part must be at least 50 cm. The set of values of $x$ and $y$ that satisfy these conditions can be described by the system of inequalities

$$
\left\{\begin{array}{c}
x \geq 50 \\
y \geq 50 \\
200-x-y \geq 50
\end{array}\right.
$$

which is also represented on the coordinate plane as a right-angled triangle with sides of 50 cm and an area of $S_{1}=\frac{1}{2} \cdot 50 \cdot 50=1250$ cm $^{2}$.

Then, the probability that the sizes of the cut parts will be suitable for flue pipes is

$$
p=\frac{S_{1}}{S}=\frac{1250}{20000}=\frac{1}{16}
$$

Remark. In solving the problem, the similarity of triangles can be used: the similarity coefficient of right-angled triangles with legs of 200 and 50, respectively, is $\frac{1}{4}$, so their areas are in the ratio $\frac{1}{16}$.

Answer. $\frac{1}{16}$.