Task 4. (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's denote the sizes of the parts into which the pipe was cut 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 XOY as a right-angled triangle with sides (Fig. 1) equal to 200 cm (Fig. 1).

![](https://cdn.mathpix.com/cropped/2024_05_06_e9b4fbbbcf148e16432dg-03.jpg?height=566&width=826&top_left_y=351&top_left_x=615)

The measure of this set can be considered the area of this triangle $S=\frac{1}{2} \cdot 200 \cdot 200=20000$ $\mathrm{cm}^{2}$.

To use the resulting parts as pipe offsets, 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 25 cm and an area of $S_{1}=\frac{1}{2} \cdot 50 \cdot 50=1250 \mathrm{~cm}^{2}$.

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

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

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

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