Question

Из двух городов навстречу друг другу выехали два автомобиля. Первый автомобиль преодолевает расстояние между этими городами за 3 часа, а второй-за 4 часа.Встретились ли автомобили, если они были в дороге 1 час? 2 часа?

Answer 1

Найдем скорости этих автомобилей, если расстояние между городами равно s
$v_1= \frac{s}{t_1} = \frac{s}{3} \\\ v_2= \frac{s}{t_2} = \frac{s}{4}$

Найдем какое расстояние в сумме они проедут за один час:
$s_1=(v_1+v_2)\tau_1=( \frac{s}{3} +\frac{s}{4})\cdot1= \frac{4s+3s}{12}= \frac{7s}{12}$
7s/12<s - значит они не встретятся через один час

Найдем какое расстояние в сумме они проедут за два часа:
$s_2=(v_1+v_2)\tau_2=( \frac{s}{3} +\frac{s}{4})\cdot2= \frac{7s\cdot2}{12}= \frac{7s}{6}$
7s/6>s - значит они встретятся через два часа

Answer 2

Problem Analysis

Two cars are traveling towards each other from two different cities. The first car takes 3 hours to travel the distance between the cities, while the second car takes 4 hours. The question is whether the cars will meet if they have been on the road for 1 hour and 2 hours.

Solution

To determine if the cars will meet, we need to compare the distances traveled by each car after 1 hour and 2 hours.

Let's assume that the distance between the two cities is D.

After 1 hour, the first car will have traveled 1/3 of the distance, and the second car will have traveled 1/4 of the distance.

After 2 hours, the first car will have traveled 2/3 of the distance, and the second car will have traveled 2/4 of the distance.

If the sum of the distances traveled by the two cars is equal to or greater than the distance between the cities, then the cars will meet.

Let's calculate the distances traveled by each car after 1 hour and 2 hours:

- After 1 hour: - First car: D/3 - Second car: D/4

- After 2 hours: - First car: 2D/3 - Second car: 2D/4 = D/2

Now, let's compare the distances traveled by the two cars after 1 hour and 2 hours:

- After 1 hour: - Sum of distances: D/3 + D/4 = (4D + 3D) / 12 = 7D / 12

- After 2 hours: - Sum of distances: 2D/3 + D/2 = (4D + 6D) / 12 = 10D / 12 = 5D / 6

If the sum of the distances traveled by the two cars is equal to or greater than the distance between the cities (D), then the cars will meet.

Therefore, the cars will meet if 7D / 12 >= D or 5D / 6 >= D.

Simplifying the inequalities:

- 7D / 12 >= D: - Multiply both sides by 12: 7D >= 12D - Subtract 7D from both sides: 0 >= 5D - Divide both sides by 5: 0 >= D - This inequality is true for any positive value of D. Therefore, the cars will meet after 1 hour.

- 5D / 6 >= D: - Multiply both sides by 6: 5D >= 6D - Subtract 5D from both sides: 0 >= D - This inequality is true for any positive value of D. Therefore, the cars will meet after 2 hours.

In conclusion, the cars will meet after both 1 hour and 2 hours of travel.

Answer

Yes, the cars will meet if they have been on the road for 1 hour and 2 hours.

Note: The above analysis assumes that the cars are traveling at a constant speed and there are no other factors that could affect their meeting.