Task 5. (20 points) A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if according to regulations, the stove should not be located closer than 75 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 $(300-x-y)$.

Obviously, the values of $x$ and $y$ can take any values from the interval $(0; 300)$. 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 300 cm (Fig. 2).

![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-36.jpg?height=514&width=828&top_left_y=474&top_left_x=614)

Fig. 2

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

$$
S=\frac{1}{2} \cdot 300 \cdot 300=45000 \mathrm{~cm}^{2}
$$

For the resulting parts to be used as tile offsets, the size of each part must be at least $75 \mathrm{~cm}$. The set of values of $x$ and $y$ that satisfy these conditions can be described by the system of inequalities

$$
\left\{\begin{array}{c}
x \geq 75 \\
y \geq 75 \\
300-x-y \geq 75
\end{array}\right.
$$

which is also represented on the coordinate plane as a right-angled triangle with sides of 75 cm and an area of $S_{1}=\frac{1}{2} \cdot 75 \cdot 75=2812.5 \mathrm{~cm}^{2}$.

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

$$
p=\frac{S_{1}}{S}=\frac{2812.5}{45000}=\frac{1}{16}
$$

Remark. In solving the problem, the similarity of triangles can be used: the similarity coefficient of right-angled triangles with legs of 300 and 75, respectively, is $\frac{1}{4}$, so their areas are in the ratio $\frac{1}{16}$.

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