Question

Из города в село, находящегося от него на расстоянии 420 км, выехали одновременно два автомобиля. Скорость первого была на 35 км/ч больше скорости второго, и поэтому он пришел к месту назначения на 2 часа раньше. Найдите скорость первого автомобиля.​

Answer 1

Пусть х км/ч - скорость первого автомобиля, тогда (х - 35) км/ч - скорость второго автомобиля. Уравнение:

420/(х-35) - 420/х = 2

420 · х - 420 · (х - 35) = 2 · х · (х - 35)

420х - 420х + 14700 = 2х² - 70х

2х² - 70х - 14700 = 0

х² - 35х - 7350 = 0

D = b² - 4ac = (-35)² - 4 · 1 · (-7350) = 1225 + 29400 = 30625

√D = √30625 = 175

х₁ = (35-175)/(2·1) = (-140)/2 = -70 (не подходит, так как < 0)

х₂ = (35+175)/(2·1) = 210/2 = 105

Ответ: 105 км/ч.

Проверка:

420 : 105 = 4 ч - время движения первого автомобиля

420 : (105 - 35) = 420 : 70 = 6 ч - время движения второго автомобиля

6 - 4 = 2 ч - разница

Answer 2

Problem Analysis

We are given that two cars, traveling from a city to a village located 420 km away, started at the same time. The speed of the first car is 35 km/h faster than the speed of the second car. As a result, the first car arrived at the destination 2 hours earlier than the second car. We need to find the speed of the first car.

Solution

Let's assume the speed of the second car is x km/h. Then, the speed of the first car would be x + 35 km/h.

We can use the formula distance = speed × time to calculate the time taken by each car to reach the destination.

For the second car: - Distance = 420 km - Speed = x km/h - Time = 420 / x hours

For the first car: - Distance = 420 km - Speed = x + 35 km/h - Time = 420 / (x + 35) hours

According to the problem, the first car arrived 2 hours earlier than the second car. So, we can set up the following equation:

420 / x - 420 / (x + 35) = 2

Now, let's solve this equation to find the value of x.

Calculation

To solve the equation 420 / x - 420 / (x + 35) = 2, we can multiply both sides of the equation by x(x + 35) to eliminate the denominators.

This gives us:

420(x + 35) - 420x = 2x(x + 35)

Simplifying the equation:

420x + 14700 - 420x = 2x^2 + 70x

Combining like terms:

14700 = 2x^2 + 70x

Rearranging the equation:

2x^2 + 70x - 14700 = 0

Now, we can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, the values of a, b, and c are:

a = 2 b = 70 c = -14700

Substituting these values into the quadratic formula:

x = (-70 ± √(70^2 - 4 * 2 * -14700)) / (2 * 2)

Simplifying the equation:

x = (-70 ± √(4900 + 117600)) / 4

x = (-70 ± √122500) / 4

x = (-70 ± 350) / 4

Now, we have two possible values for x:

1. x = (-70 + 350) / 4 = 280 / 4 = 70 2. x = (-70 - 350) / 4 = -420 / 4 = -105

Since the speed of a car cannot be negative, we can discard the second solution.

Therefore, the speed of the second car is 70 km/h.

The speed of the first car is x + 35 = 70 + 35 = 105 km/h.

Answer

The speed of the first car is 105 km/h.