Question

В 15:00 часовая и минутная стрелки на циферблате часов перпендикулярны друг другу. Через какое время после 15:00 они будут перпендикулярны друг другу в следующий раз?

Answer 1

Решение:

Определяем время по формуле, связывающей угол между двумя стрелками и позициями этих стрелок на циферблате:

2U = 60НН - 11ММ

2*1 = 60*3 - 11ММ

11ММ = 2*180

ММ = 360/11 = 32,7 мин.

Ответ: через 32,7 мин. часовая и минутная стрелка будут перпендикулярны.

Answer 2

Calculation of the Time when the Hour and Minute Hands are Perpendicular

To determine the time when the hour and minute hands on a clock are perpendicular to each other, we need to consider the relationship between the hour and minute hands and their respective speeds.

The minute hand moves 360 degrees in 60 minutes, which means it moves 6 degrees per minute. The hour hand moves 360 degrees in 12 hours, which means it moves 0.5 degrees per minute.

Let's assume that at 15:00 (3:00 PM), the hour and minute hands are perpendicular to each other. We want to find out when they will be perpendicular again.

To find the time, we need to calculate the angle between the hour and minute hands at 15:00 and then determine how long it takes for the minute hand to catch up to the hour hand by closing that angle.

At 15:00, the minute hand is pointing at the 12, while the hour hand is pointing at the 3. The angle between them is 90 degrees.

To calculate the time it takes for the minute hand to catch up to the hour hand by closing a 90-degree angle, we can set up the following equation:

0.5x = 6x - 90

Where: - x represents the time in minutes it takes for the minute hand to catch up to the hour hand. - 0.5x represents the angle covered by the hour hand in x minutes (0.5 degrees per minute). - 6x represents the angle covered by the minute hand in x minutes (6 degrees per minute). - 90 represents the initial angle between the hour and minute hands at 15:00.

Solving the equation:

0.5x = 6x - 90

5.5x = 90

x = 90 / 5.5

x ≈ 16.36 minutes

Therefore, it will take approximately 16.36 minutes for the minute hand to catch up to the hour hand and for them to be perpendicular to each other again after 15:00.

Please note that this calculation assumes a standard 12-hour analog clock and does not account for any variations or complications in clock mechanisms.