Task 5. (20 points) A four-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 1 m 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 $(400-x-y)$.

Obviously, the values of $x$ and $y$ can take any values from the interval (0; 400). 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 400 cm (Fig. 2).

![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-27.jpg?height=568&width=834&top_left_y=1578&top_left_x=608)

The measure of this set can be considered as the area of this triangle $S=\frac{1}{2} \cdot 400 \cdot 400=80000$ cm $^{2}$. For the resulting parts to be suitable as pipe offsets, the size of each part must be at least 1 m, i.e., 100 cm. The set of values of $x$ and $y$ that satisfy these conditions can be described by the system of inequalities

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

which is also represented on the coordinate plane as a right-angled triangle with sides of 100 cm and an area of $S_{1}=\frac{1}{2} \cdot 100 \cdot 100=5000$ cm $^{2}$.

Then, the probability that the sizes of the cut parts will be suitable for pipe offsets is

$$
p=\frac{S_{1}}{S}=\frac{5000}{80000}=\frac{1}{16}
$$

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