Question

Даю 15 баллов Тик-так Любопытный Кирилл обратил внимание, что в 22:50 угол между часовой и минутной стрелкой равен X градусов. Немного подумав, Кирилл понял, что такое же значение угла между часовой и минутной стрелками будет в 13 часов N минут. Найдите X. Найдите также N, если известно, что оно целое. Чему равен Х?

Answer 1

Считаем что циферблат 12 часовой

полный оборот круга 360 градусов

за час часовая стрелка проходит 360 : 12 = 30 градусов

Минутная за минуту 360 : 60 = 6 градусов

каждую минуту часовая передвигается на 30 : 60 = 1/2 градуса

теперь можно вывести формулу градусов между стрелками

градус  = |градус часовой - градус минутной| = |(номер часа + кол-во минут / 60)* 30 - кол-во минут * 6|

взяли по модулю так как градус положительный должен быть и какая стрелка какую обогнала нас не интересует

g = |(h + m/60)*30 - 6m| = |30h - m/2 - 6m| = |30h - 11m/2|

22 = 10 вечера

22: 50 = |30*10 - 11*50/2| = |300 - 275| = 25

x = 25

13 = 1 дня

13 : n = |30*1 - 11n/2|

|30*1 - 11n/2| = 25  

11n/2 = 55

n = 10

в 13 : 10 угол 25

ответ Х = 25  n = 10

Answer 2

Problem Analysis

Kirill noticed that the angle between the hour and minute hands of a clock at 22:50 is equal to X degrees. He also realized that the same angle between the hour and minute hands would occur at 13 hours N minutes. We need to find the value of X and determine the value of N, knowing that it is an integer.

Solution

To solve this problem, we need to understand the relationship between the positions of the hour and minute hands on a clock.

The hour hand moves 360 degrees in 12 hours, which means it moves 30 degrees per hour. Additionally, it moves 0.5 degrees per minute (since there are 60 minutes in an hour and 30 degrees divided by 60 minutes equals 0.5 degrees per minute).

The minute hand moves 360 degrees in 60 minutes, which means it moves 6 degrees per minute.

Let's calculate the angle between the hour and minute hands at 22:50.

The hour hand is at 22 hours, which means it has moved 22 hours * 30 degrees per hour = 660 degrees.

The minute hand is at 50 minutes, which means it has moved 50 minutes * 6 degrees per minute = 300 degrees.

The angle between the hour and minute hands is the absolute difference between their positions, so it is |660 degrees - 300 degrees| = 360 degrees.

Now, let's find the value of N when the angle between the hour and minute hands is also X degrees at 13 hours N minutes.

At 13 hours, the hour hand has moved 13 hours * 30 degrees per hour = 390 degrees.

Let's assume the minute hand has moved N minutes * 6 degrees per minute = N * 6 degrees.

The angle between the hour and minute hands is |390 degrees - (N * 6 degrees)| = X degrees.

Since we know that the angle between the hour and minute hands is the same at 22:50 and 13 hours N minutes, we can equate the two expressions for the angle:

360 degrees = |390 degrees - (N * 6 degrees)|

To solve this equation, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: 390 degrees - (N * 6 degrees) > 0 In this case, the equation becomes: 360 degrees = 390 degrees - (N * 6 degrees)

Simplifying the equation: (N * 6 degrees) = 30 degrees

Dividing both sides by 6 degrees: N = 5

Case 2: 390 degrees - (N * 6 degrees) < 0 In this case, the equation becomes: 360 degrees = -(390 degrees - (N * 6 degrees))

Simplifying the equation: (N * 6 degrees) = 390 degrees - 360 degrees

Simplifying further: (N * 6 degrees) = 30 degrees

Dividing both sides by 6 degrees: N = 5

Therefore, regardless of whether the expression inside the absolute value is positive or negative, N is equal to 5.

So, the value of X is 360 degrees and the value of N is 5.

Answer

The angle between the hour and minute hands at 22:50 is 360 degrees. Additionally, the same angle between the hour and minute hands will occur at 13 hours 5 minutes.