11. Toll Road (from 8th grade. 3 points). The cost of traveling along a section of a toll road depends on the class of the vehicle: passenger cars belong to the first class, for which the travel cost is 200 rubles, and light trucks and minivans belong to the second class, for which the cost is 300 rubles.

At the entrance to the toll booth, an automatic system is installed that measures the height of the vehicle. If the height of the vehicle is less than a certain threshold value \( h \), the system automatically classifies it as class 1; if it is higher than \( h \), it is classified as class 2. Errors are possible. For example, a low minivan may be classified as class 1 by the system, and its driver will be pleased. A low SUV may be incorrectly classified as class 2, and its driver will not be happy. He can file a claim, and the operating company will have to refund him 100 rubles.

The company's management tasked the engineers with the goal of configuring the system so that the number of errors is minimized.

For several weeks, the engineers collected data on the height of vehicles passing through the toll booth. Separately for class 1 vehicles and separately for class 2 vehicles (Fig. 5). On the x-axis, the height of the vehicle (in cm) is plotted, and on the y-axis, the average number of vehicles of such height passing through the toll booth per day. For clarity, the points are connected by a smooth line.
![](https://cdn.mathpix.com/cropped/2024_05_06_cb774fbc1caae3aa402cg-05.jpg?height=486&width=1628&top_left_y=274&top_left_x=266)

Fig. 5. Graphs - the number of class 1 and class 2 vehicles per day

Having gathered all this information and drawn the graphs, the engineers began to think about what to do next: how to determine the threshold value \( h \) so that the probability of error is the smallest possible? Solve this problem for them.
Solution. We will construct both graphs in the same coordinate system. Draw a vertical line $x=h$ through the point of intersection of the graphs. This value of $h-$ is the one we are looking for.

![](https://cdn.mathpix.com/cropped/2024_05_06_cb774fbc1caae3aa402cg-05.jpg?height=634&width=1445&top_left_y=1268&top_left_x=363)

We will prove this geometrically. Let's call the point of intersection $C$. Draw any vertical line $x=h$ (for definiteness, to the right of the point of intersection of the graphs). Introduce the notation for a few more points, as shown

![](https://cdn.mathpix.com/cropped/2024_05_06_cb774fbc1caae3aa402cg-05.jpg?height=383&width=855&top_left_y=1959&top_left_x=1026)

Fig. 3. as shown in Fig. 3.

There can be two types of errors. The first type is when the system incorrectly classifies a class 1 car as class 2. The number of such errors is represented by the area of the figure under the graph 1 to the right of the line $x=h$, that is, the area of the curvilinear triangle $G F D$. The second type of error: the system incorrectly classifies class 2 as class 1. The number of such errors is equal to the area of the curvilinear triangle $A E G$ under the graph 2 to the left of the line $x=h$. The figure composed of triangles $F G D$ and $A E G$ has the smallest area when the area of triangle $C E F$ is the smallest. This area is zero only if the line $x=h$ coincides with the line $B C$. Therefore, the value we are looking for is the abscissa of the common point of the graphs ${ }^{2}$: approximately 190 cm.

Answer: approx. 190 cm.