Question

Лыжная трасса представляет собой контур прямоугольного треугольника с разностью катетов в 2 км. Два лыжника стартовали одновременно из вершины одного острого угла. Один поехал по гипотенузе со скоростью 15 км/ч, а другой по катетам со скоростью 21 км/ч. В вершину другого острого угла оба лыжника прибыли одновременно. Определите протяжённость трассы в километрах.​

Answer 1

Ответ:

$\frac{4+10+\sqrt{116} }{3}$ км протяженность всей трассы

Объяснение:

Пусть длина одного катета x км, значит длина второго катета (х+2) км.

Гипотенуза треугольника равна:

$\sqrt{x^2+(x+2)^2}=\sqrt{2x^2+4x+4}$

Лыжник ехал по катетам (х+х+2)/21 (ч).

Лыжнитк ехал по гипотенузе $\frac{\sqrt{2x^2+4x+4} }{15}$ (ч).

Составим и решим уравнение:

$\frac{\sqrt{2x^2+4x+4} }{15}=\frac{2x+2}{21} \\ \\ 7\sqrt{2x^2+4x+4}=10(x+1)\\ \\ 49(x^{2} +4x+4)= 100(x^{2} +2x+1)\\ \\ 49x^{2} +196x+196=100x^{2} +200x+100\\ \\ 51x^{2} +4x-96=0\\ \\ D=4^2+4*51*96=140^2\\ \\ x_1=\frac{-4-140}{51*2}= -\frac{72}{51}$

$x_2=\frac{-4+140}{2*51}= \frac{4}{3}=1\frac{1}{3}$  км катет трассы

$1\frac{1}{3}+2=3\frac{1}{3}$ км второй катет трассы

$\sqrt{(\frac{4}{3} )^2+(\frac{10}{3})^2 }=\frac{\sqrt{116} }{3}$ км гипотенуза

$1\frac{1}{3}+3\frac{1}{3} +\frac{\sqrt{116} }{3}=\frac{4+10+\sqrt{116} }{3}$ км длина всей трассы

Answer 2

To determine the length of the ski track, we can use the concept of relative speed. Let's break down the problem step by step.

Given information:

- The ski track is in the shape of a right-angled triangle. - The difference between the lengths of the two legs (or catheti) is 2 km. - Two skiers start simultaneously from the top of one acute angle. - One skier travels along the hypotenuse at a speed of 15 km/h. - The other skier travels along the legs at a speed of 21 km/h. - Both skiers arrive at the top of the other acute angle at the same time.

Approach:

1. Let's assume the length of one leg of the triangle is x km. Therefore, the length of the other leg will be (x + 2) km. 2. We can calculate the time taken by each skier using the formula: time = distance / speed. 3. The time taken by the skier traveling along the hypotenuse will be equal to the time taken by the skier traveling along the legs. 4. Using the time taken and the speed, we can calculate the distances traveled by each skier. 5. Finally, we can add up the distances traveled along the legs to find the total length of the ski track.

Calculation:

Let's calculate the distances traveled by each skier.

- Skier traveling along the hypotenuse: - Time taken = Distance / Speed = x km / 15 km/h = x/15 hours - Distance traveled = Speed * Time taken = 15 km/h * (x/15) hours = x km

- Skier traveling along the legs: - Time taken = Distance / Speed = (x + 2) km / 21 km/h = (x + 2)/21 hours - Distance traveled = Speed * Time taken = 21 km/h * ((x + 2)/21) hours = (x + 2) km

Since both skiers arrive at the top of the other acute angle at the same time, the distances traveled by each skier are equal. Therefore, we can equate the distances traveled:

x = x + 2

Solving this equation, we find that x = -2. However, since distance cannot be negative, this solution is not valid.

Therefore, there is no valid solution for the given problem. Please double-check the information provided or the problem statement itself.

If you have any further questions, feel free to ask!