Question

Скорость одного поезда на 10 км/ч больше, чем скорость другого, и поэтому путь в 80 км он проходит на 16 мину быстрее. Найдите скорость каждого поезда.

Answer 1

Ответ:   50 км/ч  и  60 км/ч

Решение:

Пусть скорость первого поезда равна  x км/ч.

Тогда скорость второго равна  x + 10 км/ч.

Время первого поезда - $\frac{80}{x}$ часов.

Тогда время второго - $\frac{80}{x + 10}$ .

Но также мы знаем, что время второго поезда меньше на 16 минут, то есть $\frac{16}{60} = \frac{4}{15}$ часа. Можно составить уравнение и решить его:

   $\frac{80}{x + 10} + \frac{4}{15} = \frac{80}{x}\\\frac{80}{x} - \frac{80}{x + 10} = \frac{4}{15} \\\frac{80*(x+10) - 80*x}{x*(x+10)} = \frac{4}{15} \\\frac{800}{x^2+10x} = \frac{4}{15} \\ 800*15 = 4*(x^2+10x)\\200*15 = x^2 + 10x\\x^2 + 10x - 3000 = 0.\\x(1) = -60.\\x(2) = 50.$

Очевидно, что первый корень (-60) не подходит, так что скорость первого поезда равна 50 км/ч, а второго - 50 + 10 = 60 км/ч.

Ура! Задача решена! Удачи!

Answer 2

Problem Analysis

We are given that the speed of one train is 10 km/h greater than the speed of another train. The difference in their speeds allows the first train to cover a distance of 80 km in 16 minutes less time than the second train. We need to find the speed of each train.

Solution

Let's assume the speed of the slower train is x km/h. Therefore, the speed of the faster train is x + 10 km/h.

We know that time is equal to distance divided by speed. The time taken by the slower train to cover a distance of 80 km is given by 80 / x hours. The time taken by the faster train to cover the same distance is given by 80 / (x + 10) hours.

According to the problem, the faster train takes 16 minutes less time than the slower train. Since 1 hour is equal to 60 minutes, we can convert 16 minutes to hours by dividing it by 60: 16 / 60 = 4/15 hours.

Therefore, we can set up the equation:

80 / x - 80 / (x + 10) = 4/15

To solve this equation, we can multiply both sides by 15x(x + 10) to eliminate the denominators:

15x(x + 10) * (80 / x - 80 / (x + 10)) = 15x(x + 10) * (4/15)

Simplifying the equation:

80(x + 10) - 80x = 4x(x + 10)

Simplifying further:

80x + 800 - 80x = 4x^2 + 40x

Simplifying again:

800 = 4x^2 + 40x

Rearranging the equation:

4x^2 + 40x - 800 = 0

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

Solving the Quadratic Equation

We can solve the quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 4, b = 40, and c = -800.

Substituting the values into the quadratic formula:

x = (-40 ± √(40^2 - 4 * 4 * -800)) / (2 * 4)

Simplifying:

x = (-40 ± √(1600 + 12800)) / 8

x = (-40 ± √14400) / 8

x = (-40 ± 120) / 8

Now we have two possible values for x:

x = (-40 + 120) / 8 = 80 / 8 = 10

x = (-40 - 120) / 8 = -160 / 8 = -20

Since speed cannot be negative, we discard the negative value of x.

Therefore, the speed of the slower train is 10 km/h.

The speed of the faster train is 10 + 10 = 20 km/h.

Answer

The speed of the slower train is 10 km/h, and the speed of the faster train is 20 km/h.