Question

Один из лыжников прошёл расстояние в 13 км на 13 мин быстрее, чем другой. Найдите скорость каждого лыжника, зная, что один из них двигался со скоростью, на 2 км/ч большей, чем другой.

Answer 1

Х км/ч - скорость одного лыжника
х+2 км/ч - скорость другого лыжника
13/х ч - время одного лыжника
13/(х+2) ч время другого лыжника
Разница во времени 13/60 ч.
$\frac{13}{x}- \frac{13}{x+2}= \frac{13}{60}$
Преобразовав уравнение получаем: $x^{2} +2x-120=0$/
Корни уравнения -12 и 10. Первый не подходит.
Скорость одного лыжника 10 км/ч
Скорость другого лыжника 12 км/ч

Answer 2

Problem Analysis

We are given that one skier traveled a distance of 13 km in 13 minutes faster than the other skier. We need to find the speed of each skier, knowing that one of them was moving at a speed 2 km/h faster than the other.

Solution

Let's assume the speed of the slower skier is x km/h. Therefore, the speed of the faster skier would be x + 2 km/h.

We can use the formula speed = distance / time to find the speed of each skier.

For the slower skier: - Distance = 13 km - Time = t minutes (unknown)

For the faster skier: - Distance = 13 km - Time = t - 13 minutes (since the faster skier took 13 minutes less)

Using the formula, we can set up the following equations:

Speed of slower skier: x = 13 / t km/h

Speed of faster skier: x + 2 = 13 / (t - 13) km/h

To solve these equations, we need to find the value of t.

Let's solve the equations step by step:

1. From the equation for the speed of the slower skier, we have: x = 13 / t

2. From the equation for the speed of the faster skier, we have: x + 2 = 13 / (t - 13)

3. Substitute the value of x from the first equation into the second equation: 13 / t + 2 = 13 / (t - 13)

4. Cross-multiply to eliminate the denominators: 13(t - 13) = 13t + 2t(t - 13)

5. Simplify the equation: 13t - 169 = 13t + 2t^2 - 26t

6. Rearrange the equation: 2t^2 - 26t - 169 = 0

Now, we can solve this quadratic equation to find the value of t.

Using the quadratic formula t = (-b ± sqrt(b^2 - 4ac)) / (2a), where a = 2, b = -26, and c = -169, we can calculate the value of t.

Let's calculate the value of t using the quadratic formula:

t = (-(-26) ± sqrt((-26)^2 - 4 * 2 * (-169))) / (2 * 2)

Simplifying further:

t = (26 ± sqrt(676 + 1352)) / 4

t = (26 ± sqrt(2028)) / 4

t = (26 ± sqrt(4 * 507)) / 4

t = (26 ± 2sqrt(507)) / 4

t = 13 ± sqrt(507) / 2

Therefore, the possible values of t are 13 + sqrt(507) / 2 and 13 - sqrt(507) / 2.

Now, let's calculate the speeds of the skiers using these values of t.

For the slower skier: x = 13 / t km/h

For the faster skier: x + 2 = 13 / (t - 13) km/h

Let's calculate the speeds using both values of t.

For t = 13 + sqrt(507) / 2: - Speed of slower skier: x = 13 / (13 + sqrt(507) / 2) km/h - Speed of faster skier: x + 2 = 13 / (13 + sqrt(507) / 2 - 13) km/h

For t = 13 - sqrt(507) / 2: - Speed of slower skier: x = 13 / (13 - sqrt(507) / 2) km/h - Speed of faster skier: x + 2 = 13 / (13 - sqrt(507) / 2 - 13) km/h

Let's calculate these speeds:

For t = 13 + sqrt(507) / 2: - Speed of slower skier: x = 13 / (13 + sqrt(507) / 2) km/h - Speed of faster skier: x + 2 = 13 / (13 + sqrt(507) / 2 - 13) km/h

For t = 13 - sqrt(507) / 2: - Speed of slower skier: x = 13 / (13 - sqrt(507) / 2) km/h - Speed of faster skier: x + 2 = 13 / (13 - sqrt(507) / 2 - 13) km/h

Please note that the above calculations are based on the given information and the equations derived from it.