Question

. Из пункта A в пункт B одновременно выехали два автомобиля. Первый проехал с постоянной скоростью весь путь. Второй проехал первую половину пути со скоростью 30 км/ч, а вторую половину пути — со скоростью, на 20 км/ч большей скорости первого, в результате чего прибыл в пункт В одновременно с первым автомобилем. Найдите скорость первого автомобиля. Ответ дайте в км/ч. 

Answer 1

Problem Analysis

We are given that two cars travel from point A to point B simultaneously. The first car travels the entire distance at a constant speed, while the second car travels the first half of the distance at a speed of 30 km/h and the second half at a speed 20 km/h faster than the first car. We need to find the speed of the first car.

Solution

Let's assume the distance between point A and point B is d.

The time taken by the first car to travel the distance is given by: t1 = d / v1, where v1 is the speed of the first car.

The time taken by the second car to travel the first half of the distance is given by: t2 = (d / 2) / v2, where v2 is the speed of the second car for the first half of the distance.

The time taken by the second car to travel the second half of the distance is given by: t3 = (d / 2) / (v1 + 20), where v1 + 20 is the speed of the second car for the second half of the distance.

Since both cars arrived at point B simultaneously, we can equate the total time taken by both cars: t1 = t2 + t3.

Substituting the values, we get: d / v1 = (d / 2) / v2 + (d / 2) / (v1 + 20).

Simplifying the equation, we get: v1 = (2 * v2 * (v1 + 20)) / (v2 + v1 + 20).

Now we can substitute the given value of v2 = 30 km/h and solve for v1.

Calculation

Let's calculate the speed of the first car using the given values.

Substituting v2 = 30 km/h into the equation, we get: v1 = (2 * 30 * (v1 + 20)) / (30 + v1 + 20).

Simplifying the equation, we get: v1 = (60 * (v1 + 20)) / (50 + v1).

Cross-multiplying, we get: v1 * (50 + v1) = 60 * (v1 + 20).

Expanding and simplifying, we get: 50v1 + v1^2 = 60v1 + 1200.

Rearranging the equation, we get: v1^2 - 10v1 - 1200 = 0.

Factoring the quadratic equation, we get: v1 = -40 or v1 = 30.

Since the speed cannot be negative, the speed of the first car is 30 km/h.

Answer

The speed of the first car is 30 km/h.