Question

На схеме указаны дороги между четырьмя населенными пунктами и отмечены длины этих дорог. Длина одной дороги неизвестна. Зато известно, что самый короткий путь между двумя наиболее удаленными друг от друга населенными пунктами составляет 70 км. Найдите длину неизвестной дороги.

Answer 1

Если ни один из известных путей между наиболее удаленными друг от друга населенными пунктами не равен 70, то задача решается вычитанием из 70 наиболее короткого пути от наиболее удаленного населенного пункта до точки от которой дорога неизвестна.

Answer 2

Problem Analysis

We are given a diagram showing roads between four towns, with the lengths of these roads marked. The length of one road is unknown. However, we know that the shortest path between the two most distant towns is 70 km. We need to find the length of the unknown road.

Solution

To find the length of the unknown road, we can use the concept of the triangle inequality. According to the triangle inequality, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In this case, we can consider the four towns as the vertices of a quadrilateral, and the roads as the sides of the quadrilateral. Let's label the towns as A, B, C, and D, and the lengths of the roads as a, b, c, and x (unknown).

We are given that the shortest path between the two most distant towns (let's say A and C) is 70 km. This means that the sum of the lengths of the two known roads (a and c) must be greater than the length of the unknown road (x).

Mathematically, we can express this as:

a + c > x (Equation 1)

Similarly, we can consider the shortest path between the other two most distant towns (B and D). Let's say the lengths of the known roads are b and x (unknown). Again, the sum of the lengths of the two known roads (b and x) must be greater than the length of the unknown road (c).

Mathematically, we can express this as:

b + x > c (Equation 2)

Now, we have two inequalities (Equation 1 and Equation 2) involving the unknown road length (x) and the known road lengths (a, b, and c). We can solve these inequalities to find the possible range of values for the unknown road length.

Let's solve these inequalities:

From Equation 1, we have:

a + c > x

From Equation 2, we have:

b + x > c

Rearranging Equation 1, we get:

x < a + c (Equation 3)

Rearranging Equation 2, we get:

x > c - b (Equation 4)

Combining Equations 3 and 4, we can conclude that:

c - b < x < a + c

Therefore, the length of the unknown road (x) must be greater than (c - b) and less than (a + c).

Let's summarize the solution:

- The length of the unknown road (x) must satisfy the inequality: c - b < x < a + c.

Note: The specific values of a, b, and c are not provided in the question, so we cannot determine the exact length of the unknown road. We can only provide the range of possible values based on the given information.

I hope this helps! Let me know if you have any further questions.