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"Youth and Elections" - The development of political legal awareness among young people: Youth and elections. Development of political legal awareness in schools and secondary specialized institutions: A set of measures to attract youth to the elections. Why don't we vote? Development of political legal consciousness in preschool educational institutions:
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"Signs of divisibility of natural numbers" - Relevance. Pascal sign. Sign of divisibility of numbers by 6. Sign of divisibility of numbers by 8. Sign of divisibility of numbers by 27. Sign of divisibility of numbers by 19. Sign of divisibility of numbers by 13. Identify signs of divisibility. How to learn to calculate quickly and correctly. Sign of divisibility of numbers by 25. Sign of divisibility of numbers by 23.
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"Competition in mathematics for schoolchildren" - Mathematical terms. The part of a line that connects two points. Students' knowledge. Contest of funny mathematicians. Task. A ray that bisects an angle. All corners are straight. Time interval. Contest. The most attractive. Speed. Radius. Getting ready for winter. Jumping dragonfly. Figure. Game with spectators. The sum of the angles of a triangle.
Total in the topic 23687 presentations
From point A of the circular track, the length of which is 75 km, two cars started simultaneously in the same direction. The speed of the first car is 89 km/h, the speed of the second car is 59 km/h. In how many minutes after the start will the first car be ahead of the second by exactly one lap?
The solution of the problem
This lesson shows how, using the physical formula for determining the time in uniform motion: , make a proportion to determine the time when one car overtakes another in a circle. When solving the problem, a clear sequence of actions is indicated for solving such problems: we introduce a specific designation for what we want to find, we write down the time it takes for one and the second car to overcome a certain number of laps, given that this time is the same value - we equate the resulting equalities . The solution is finding an unknown quantity in a linear equation. To get the results, be sure to remember to substitute the number of laps obtained in the formula for determining the time.
The solution of this problem is recommended for students of the 7th grade when studying the topic “Mathematical language. Mathematical model "(Linear equation with one variable"). When preparing for the OGE, the lesson is recommended when repeating the topic “Mathematical language. Mathematical model".
Sections: Mathematics
The article discusses tasks to help students: to develop the skills of solving text problems in preparation for the Unified State Examination, when learning to solve problems for compiling a mathematical model of real situations in all parallels of the primary and high schools. It presents tasks: for movement in a circle; to find the length of a moving object; to find the average speed.
I. Problems for motion in a circle.
Circumferential tasks proved to be difficult for many students. They are solved in almost the same way as ordinary problems for movement. They also use the formula . But there is a point to which we pay attention.
Task 1. A cyclist left point A of the circular track, and after 30 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 30 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 30 km. Give your answer in km/h.
Solution. The speeds of the participants will be taken as X km/h and y km/h. For the first time, the motorcyclist overtook the cyclist 10 minutes later, that is, one hour after the start. Up to this point, the cyclist has been on the road for 40 minutes, that is, hours. The participants in the movement have traveled the same distance, that is, y = x. Let's put the data in the table.
Table 1
The motorcyclist then overtook the cyclist a second time. This happened 30 minutes later, that is, one hour after the first overtaking. What distances did they travel? The motorcyclist overtook the cyclist. And that means he drove one lap more. That's the moment
to which you need to pay attention. One circle is the length of the track, It is equal to 30 km. Let's create another table.
table 2
We get the second equation: y - x = 30. We have a system of equations: 
In the answer, we indicate the speed of the motorcyclist.
Answer: 80 km/h.
Tasks (independently).
I.1.1. A cyclist left point “A” of the circular track, and after 40 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 36 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 36 km. Give your answer in km/h.
I.1. 2. A cyclist left point “A” of the circular track, and after 30 minutes a motorcyclist followed him. 8 minutes after departure, he caught up with the cyclist for the first time, and 12 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 15 km. Give your answer in km/h.
I.1. 3. A cyclist left point “A” of the circular track, and after 50 minutes a motorcyclist followed him. 10 minutes after departure, he caught up with the cyclist for the first time, and 18 minutes after that, he caught up with him for the second time. Find the speed of the motorcyclist if the length of the track is 15 km. Give your answer in km/h.
Two motorcyclists start simultaneously in the same direction from two diametrically opposite points of a circular track, the length of which is 20 km. In how many minutes will the motorcyclists catch up for the first time if the speed of one of them is 15 km/h more than the speed of the other?
Solution.
Picture 1
With a simultaneous start, the rider who started from “A” drove half a lap more, who started from “B”. That is 10 km. When two motorcyclists move in the same direction, the removal speed is v = -. According to the condition of the problem, v= 15 km/h = km/min = km/min is the removal speed. We find the time after which the motorcyclists catch up for the first time.
10:= 40(min).
Answer: 40 min.
Tasks (independently).
I.2.1. Two motorcyclists start simultaneously in the same direction from two diametrically opposite points of a circular track, the length of which is 27 km. In how many minutes will the motorcyclists catch up for the first time if the speed of one of them is 27 km/h more than the speed of the other?
I.2.2. Two motorcyclists start simultaneously in the same direction from two diametrically opposite points of a circular track, the length of which is 6 km. In how many minutes will the motorcyclists catch up for the first time if the speed of one of them is 9 km/h more than the speed of the other?
From one point of the circular track, the length of which is 8 km, two cars started simultaneously in the same direction. The speed of the first car is 89 km/h, and 16 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.
Solution.
x km/h is the speed of the second car.
(89 - x) km / h - removal speed.
8 km - the length of the circular track.
The equation.
(89 - x) = 8,
89 - x \u003d 2 15,
Answer: 59 km/h
Tasks (independently).
I.3.1. From one point of the circular track, the length of which is 12 km, two cars started simultaneously in the same direction. The speed of the first car is 103 km/h, and 48 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.
I.3.2. From one point of the circular track, the length of which is 6 km, two cars started simultaneously in the same direction. The speed of the first car is 114 km/h, and 9 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.
I.3.3. From one point of the circular track, the length of which is 20 km, two cars started simultaneously in the same direction. The speed of the first car is 105 km/h, and 48 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.
I.3.4. From one point of the circular track, the length of which is 9 km, two cars started simultaneously in the same direction. The speed of the first car is 93 km/h, and 15 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.
The clock with hands shows 8:00. After how many minutes will the minute hand align with the hour hand for the fourth time?
Solution. We assume that we do not solve the problem experimentally.
In one hour, the minute hand goes one circle, and the hour part of the circle. Let their speeds be 1 (laps per hour) and 
Start - at 8.00. Find the time it takes for the minute hand to overtake the hour hand for the first time.
The minute hand will go further, so we get the equation
So, for the first time, the arrows will line up through
Let the arrows line up for the second time after time z. The minute hand will travel a distance of 1 z, and the hour hand will travel one more circle. Let's write the equation:
Solving it, we get that .
So, through the arrows they will line up for the second time, another through - for the third, and even through - for the fourth time.
Therefore, if the start was at 8.00, then for the fourth time the arrows will line up through
4h = 60 * 4 min = 240 min.
Answer: 240 minutes.
Tasks (independently).
I.4.1. Clock with hands shows 4 hours 45 minutes. After how many minutes will the minute hand align with the hour hand for the seventh time?
I.4.2. The clock with hands shows exactly 2 o'clock. In how many minutes will the minute hand align with the hour hand for the tenth time?
I.4.3. The clock with hands shows 8 hours 20 minutes. After how many minutes will the minute hand align with the hour hand for the fourth time? fourth
II. Problems to find the length of a moving object.
A train moving at a uniform speed of 80 km/h passes a roadside post in 36 seconds. Find the length of the train in meters.
Solution. Since the speed of the train is indicated in hours, we will convert seconds into hours.
1) 36 sec = 
2) find the length of the train in kilometers.
80 
Answer: 800m.
Tasks (independently).
II.2. The train, moving uniformly at a speed of 60 km/h, passes a roadside post in 69 s. Find the length of the train in meters. Answer: 1150m.
II.3. A train moving uniformly at a speed of 60 km/h passes a forest belt 200 m long in 1 min 21 s. Find the length of the train in meters. Answer: 1150m.
III. Tasks for medium speed.
In a math exam, you may encounter the problem of finding the average speed. It must be remembered that the average speed is not equal to the arithmetic mean of speeds. The average speed is found by a special formula:
If there were two sections of the path, then 
.
The distance between the two villages is 18 km. The cyclist traveled from one village to another for 2 hours and returned along the same road for 3 hours. What is the average speed of the cyclist for the entire journey?
Solution:
2 hours + 3 hours = 5 hours - spent on the whole movement,
.A tourist walked at a speed of 4 km/h, then exactly the same time at a speed of 5 km/h. What is the average travel speed for the entire journey?
Let the tourist walk t h at a speed of 4 km/h and t h at a speed of 5 km/h. Then in 2t h he traveled 4t + 5t = 9t (km). The average speed of a tourist is = 4.5 (km/h).
Answer: 4.5 km/h.
We notice that the average speed of the tourist turned out to be equal to the arithmetic mean of these two speeds. It can be seen that if the time of movement on two sections of the path is the same, then the average speed of movement is equal to the arithmetic mean of the two given speeds. To do this, we solve the same problem in a general form.
The tourist walked at a speed of km / h, then exactly the same time at a speed of km / h. What is the average travel speed for the entire journey?
Let the tourist walk t h at a speed of km/h and t h at a speed of km/h. Then in 2t hours he traveled t + t = t (km). The average travel speed of a tourist is
= (km/h).The car covered some distance uphill at a speed of 42 km/h, and downhill at a speed of 56 km/h.
.
The average speed of movement is 2 s: (km/h).
Answer: 48 km/h.
A car covered some distance uphill at a speed of km/h, and downhill at a speed of km/h.
What is the average speed of the car for the entire journey?
Let the length of the path segment be equal to s km. Then the car traveled 2 s km in both directions, spending the whole way 
.
The average movement speed is 2 s: 
(km/h).
Answer: km/h.
Consider a problem in which the average speed is given, and one of the speeds needs to be determined. Equation required.
A cyclist was traveling uphill at a speed of 10 km/h, and downhill at some other constant speed. As he calculated, the average speed of movement was equal to 12 km / h.
.III.2. Half the time spent on the road, the car was traveling at a speed of 60 km/h, and the second half of the time - at a speed of 46 km/h. Find the average speed of the car for the entire journey.
III.3. On the way from one village to another, the car walked for some time at a speed of 60 km/h, then for exactly the same time at a speed of 40 km/h, then for exactly the same time at a speed equal to the average speed on the first two sections of the journey . What is the average speed for the entire journey from one village to another?
III.4. A cyclist travels from home to work at an average speed of 10 km/h and back at an average speed of 15 km/h because the road is slightly downhill. Find the average speed of the cyclist all the way from home to work and back.
III.5. The car traveled from point A to point B empty at a constant speed, and returned along the same road with a load at a speed of 60 km/h. At what speed did he travel empty if the average speed was 70 km/h?.
III.6. The car drove the first 100 km at a speed of 50 km/h, the next 120 km at a speed of 90 km/h, and then 120 km at a speed of 100 km/h. Find the average speed of the car for the entire journey.
III.7. The car drove the first 100 km at a speed of 50 km/h, the next 140 km at a speed of 80 km/h, and then 150 km at a speed of 120 km/h. Find the average speed of the car for the entire journey.
III.8. The car drove the first 150 km at a speed of 50 km/h, the next 130 km at a speed of 60 km/h, and then 120 km at a speed of 80 km/h. Find the average speed of the car for the entire journey.
III. 9. The car drove for the first 140 km at a speed of 70 km/h, the next 120 km at a speed of 80 km/h, and then 180 km at a speed of 120 km/h. Find the average speed of the car for the entire journey.
