Question

Из пунктов А и В навстречу друг другу выехали одновременно два автобуса. Первый, имея вдвое большую скорость, проехал весь путь на 2 ч быстрее второго. На сколько минут позже произошла бы их встреча, если бы скорость первого уменьшилась до скорости второго автобуса?

Answer 1

на сорок  минут позже

Answer 2

Problem Analysis

We have two buses, A and B, that start from different points and travel towards each other. Bus A has twice the speed of Bus B. We need to determine how many minutes later their meeting would occur if the speed of Bus A is reduced to the speed of Bus B.

Solution

Let's assume that the distance between the two points where the buses start is d.

We know that the time taken by Bus A to cover the distance d is 2 hours less than the time taken by Bus B.

Let's denote the speed of Bus A as v_a and the speed of Bus B as v_b.

We can set up the following equations based on the given information:

1. Time taken by Bus A to cover distance d: d/v_a 2. Time taken by Bus B to cover distance d: d/v_b 3. The time taken by Bus A is 2 hours less than the time taken by Bus B: d/v_a = d/v_b - 2

To find the time difference in minutes, we need to convert the 2-hour difference to minutes. Since 1 hour is equal to 60 minutes, 2 hours is equal to 120 minutes.

Now, let's solve the equations to find the time difference in minutes.

Calculation

We have the equation: d/v_a = d/v_b - 2

To find the time difference in minutes, we need to solve for d.

Rearranging the equation, we get: d/v_a - d/v_b = -2

Multiplying both sides of the equation by v_a * v_b, we get: d * v_b - d * v_a = -2 * v_a * v_b

Factoring out d, we get: d * (v_b - v_a) = -2 * v_a * v_b

Dividing both sides of the equation by (v_b - v_a), we get: d = -2 * v_a * v_b / (v_b - v_a)

Now, we can substitute the value of d into the equation d/v_a to find the time taken by Bus A.

The time difference in minutes will be the difference between the time taken by Bus A and the time taken by Bus B, multiplied by 60 to convert it to minutes.

Let's calculate the time difference.

Calculation Continued

Substituting the value of d into the equation d/v_a, we get: (-2 * v_a * v_b / (v_b - v_a)) / v_a

Simplifying the expression, we get: -2 * v_b / (v_b - v_a)

The time difference in minutes will be: (-2 * v_b / (v_b - v_a) - d/v_b) * 60

Now, let's substitute the value of v_a = 2 * v_b into the equation and calculate the time difference.

Calculation Continued

Substituting v_a = 2 * v_b into the equation, we get: (-2 * v_b / (v_b - 2 * v_b) - d/v_b) * 60

Simplifying the expression, we get: (-2 * v_b / (-v_b) - d/v_b) * 60

Further simplifying the expression, we get: (2 - d/v_b) * 60

Now, let's substitute the value of d = v_a * t_a into the equation, where t_a is the time taken by Bus A.

Calculation Continued

Substituting d = v_a * t_a into the equation, we get: (2 - (v_a * t_a)/v_b) * 60

Simplifying the expression, we get: (2 - 2 * t_a) * 60

Finally, let's substitute the value of t_a = d/v_a into the equation and calculate the time difference in minutes.

Calculation Continued

Substituting t_a = d/v_a into the equation, we get: (2 - 2 * (d/v_a)) * 60

Simplifying the expression, we get: (2 - 2 * (d/(2 * v_b))) * 60

Further simplifying the expression, we get: (2 - d/v_b) * 60

Now, let's substitute the value of d = -2 * v_a * v_b / (v_b - v_a) into the equation and calculate the time difference in minutes.

Calculation Continued

Substituting d = -2 * v_a * v_b / (v_b - v_a) into the equation, we get: (2 - (-2 * v_a * v_b / (v_b - v_a))/v_b) * 60

Simplifying the expression, we get: (2 + 2 * v_a / (v_b - v_a)) * 60

Now, let's substitute the value of v_a = 2 * v_b into the equation and calculate the time difference in minutes.

Calculation Continued

Substituting v_a = 2 * v_b into the equation, we get: (2 + 2 * (2 * v_b) / (v_b - 2 * v_b)) * 60

Simplifying the expression, we get: (2 + 4 * v_b / (-v_b)) * 60

Further simplifying the expression, we get: (2 - 4) * 60

Finally, let's calculate the time difference in minutes.

Calculation Continued

Calculating the expression, we get: -2 * 60 = -120

Therefore, if the speed of the first bus is reduced to the speed of the second bus, their meeting would occur 120 minutes (or 2 hours) later.

Answer

If the speed of the first bus is reduced to the speed of the second bus, their meeting would occur 120 minutes later.

Note: The above calculation assumes that the buses travel at constant speeds throughout the journey and that there are no other factors affecting their meeting time.