Task 5. (20 points) A meter-long 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 25 cm to the main gas pipe.

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# Solution.

Let the sizes of the parts into which the pipe was cut be denoted as $x, y$, and $(100-x-y)$.

Obviously, the values of $x$ and $y$ can take any values from the interval (0; 100). Then, the set of all possible combinations $(x; y)$ can be represented on the coordinate plane $OXY$ as a right-angled triangle with sides equal to 100 cm (Fig. 2).

![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-22.jpg?height=575&width=809&top_left_y=1163&top_left_x=632)

The measure of this set can be considered as the area of this triangle $S=\frac{1}{2} \cdot 100 \cdot 100=5000$ cm².

For the resulting parts to be suitable as flue outlets, the size of each part must be at least 25 cm. The set of values of $x$ and $y$ that satisfy these conditions can be described by the system of inequalities

$$
\left\{\begin{array}{l}
x \geq 25 \\
y \geq 25 \\
100-x-y \geq 25
\end{array}\right.
$$

which is also represented on the coordinate plane as a right-angled triangle with sides of 25 cm and an area of $S_{1}=\frac{1}{2} \cdot 25 \cdot 25=312.5 \text{ cm}^2$.

Thus, the probability that the sizes of the cut parts will be suitable for flue outlets is $p=\frac{S_{1}}{S}=\frac{312.5}{5000}=\frac{1}{16}$.

Remark. In solving the problem, the similarity of triangles can be used:

the similarity coefficient of the right-angled triangles with legs of 100 and 25, respectively, is $\frac{1}{4}$, so their areas are in the ratio $\frac{1}{16}$.