Show that the minute hand of a watch gains 5 degree and 30' on the hour hand in a minute

A cow is tied to a post by a rope 30m long. If the cow moves along the circumference of a circle always keeping the rope tight,how far will it have gone when the rope has traced an angle of 105°.

Class 11th

on one minute, the minute hand advances 1/60 * 360 = 6 degrees

the hour hand moves 1/12 as fast, so it advances 0.5 degrees.
hence 6 - 0.5
= 5.5 = 5 30

To show that the minute hand of a watch gains 5 degrees and 30 minutes on the hour hand in a minute, we can break it down into steps:

Step 1: Calculate the rate at which the minute hand moves.
The minute hand completes a full circle (360 degrees) in 60 minutes, so its rate of movement can be calculated as:
Rate of minute hand = 360 degrees / 60 minutes = 6 degrees per minute

Step 2: Calculate the rate at which the hour hand moves.
The hour hand completes a full circle (360 degrees) in 12 hours, which is equivalent to 720 minutes. Therefore, its rate of movement can be calculated as:
Rate of hour hand = 360 degrees / 720 minutes = 0.5 degrees per minute

Step 3: Calculate the difference in rates between the minute and hour hands.
The difference in rates between the minute and hour hands can be calculated as:
Difference in rates = Rate of minute hand - Rate of hour hand
Difference in rates = 6 degrees per minute - 0.5 degrees per minute
Difference in rates = 5.5 degrees per minute

Step 4: Calculate the time it takes for the minute hand to gain 5 degrees and 30 minutes on the hour hand.
To calculate the time it takes for the minute hand to gain 5 degrees and 30 minutes on the hour hand, we can use the formula:
Time = Difference in angle / Difference in rates

Substituting the values:
Time = (5 degrees + 30 minutes) / 5.5 degrees per minute

To convert 30 minutes to degrees, we know that 1 degree is equivalent to 60 minutes, so 30 minutes is equal to 0.5 degrees.
Time = (5 degrees + 0.5 degrees) / 5.5 degrees per minute
Time = 5.5 degrees / 5.5 degrees per minute
Time = 1 minute

Hence, it takes 1 minute for the minute hand on a watch to gain 5 degrees and 30 minutes on the hour hand.

on one minute, the minute hand advances 1/60 * 360 = 6 degrees

the hour hand moves 1/12 as fast, so it advances 0.5 degrees.

ta-daa.