Comprehensive guide to calculating the angle between the hour and minute hands of a clock for CBSE Class 11 Applied Mathematics, covering angular speeds, relative movement, and step-by-step problem-solving techniques.
A clock is a circular dial of 360 degrees. The minute hand moves 360 degrees in 60 minutes, which means its speed is 6 degrees per minute. The hour hand moves 360 degrees in 12 hours (720 minutes), which means its speed is 0.5 degrees per minute. The angle between the hands depends on the relative position of both hands at any given time (H:M).
• Identify the values: H = 4 and M = 20.
• Apply the formula: Angle = |(30 * 4) - (5.5 * 20)|.
• Calculate values: 30 * 4 = 120 and 5.5 * 20 = 110.
• Find the absolute difference: |120 - 110| = 10 degrees.
Answer: The angle between the hands at 4:20 is 10 degrees.
• Identify the values: H = 8 and M = 15.
• Apply the formula: Angle = |(30 * 8) - (5.5 * 15)|.
• Calculate values: 30 * 8 = 240 and 5.5 * 15 = 82.5.
• Find the absolute difference: |240 - 82.5| = 157.5 degrees.
Answer: The angle between the hands at 8:15 is 157.5 degrees.
The formula |(30 * H) - (5.5 * M)| provides the smaller interior angle. If the question specifically asks for the reflex angle (the angle greater than 180 degrees), subtract the calculated angle from 360 degrees.
• Calculate the interior angle: |(30 * 3) - (5.5 * 0)| = |90 - 0| = 90 degrees.
• Calculate the reflex angle by subtracting from 360 degrees.
• 360 - 90 = 270 degrees.
Answer: The reflex angle between the hands at 3:00 is 270 degrees.