What is the circular (radian) measure of the angle between the hour hand and the minute hand of a clock at exactly 3:00 p.m.?

Introduction / Context:
This question involves the relationship between degree measure and radian measure on a clock face. At exactly 3:00 p.m., the hour hand and minute hand form a specific angle, and you are asked to express that angle in circular measure (radians). Understanding the conversion between degrees and radians is fundamental for higher mathematics, trigonometry, and aptitude problems involving angular motion.

Given Data / Assumptions:

• At 3:00 p.m., the minute hand is at 12 and the hour hand is at 3 on a 12-hour clock.

• The clock face is a full circle, 360 degrees, corresponding to 2π radians.

• The numbers 12, 3, 6, and 9 on the clock divide the circle into four equal quadrants.

• We must find the angle between the hands in radians.

Concept / Approach:
At 3:00 p.m., the minute hand points to 12 and the hour hand points to 3. The angle between them is clearly one-quarter of the full circle. One-quarter of 360 degrees is 90 degrees. We then convert this 90 degree angle into radians using the fact that 180 degrees corresponds to π radians. Thus, 90 degrees corresponds to π/2 radians. This direct reasoning makes the problem simple and quick to solve.

Step-by-Step Solution:
Step 1: At 3:00 p.m., the minute hand is at 12, and the hour hand is at 3. Step 2: The positions 12, 3, 6, and 9 divide the clock into four equal parts. Step 3: Each part represents a right angle, which is 360 / 4 = 90 degrees. Step 4: Therefore, the angle between the hands at 3:00 p.m. is 90 degrees. Step 5: Use the conversion: 180 degrees = π radians. Step 6: Hence, 90 degrees = (90 / 180) * π = π/2 radians. Step 7: Thus, the circular measure of the angle between the hands at 3:00 p.m. is π/2 radians.

Verification / Alternative check:
We can also reason that each hour marking on a 12-hour clock corresponds to 30 degrees (since 360 / 12 = 30). From 12 to 3 is 3 hour marks, so the angle between them is 3 * 30 = 90 degrees. Converting 90 degrees to radians again gives π/2 radians. This independently confirms the same result.

Why Other Options Are Wrong:
• π/4 radians: This corresponds to 45 degrees, which is half of the right angle and would occur at different times, not at exactly 3:00 p.m.
• π/3 radians: This is 60 degrees, smaller than the angle between 12 and 3 on the clock.
• 5π/12 radians: This equals 75 degrees, which also does not correspond to the standard quarter-circle separation at 3:00 p.m.

Common Pitfalls:
Students sometimes confuse degree and radian measures or apply the conversion factor incorrectly. Another error is misidentifying the angle between the hands at 3:00 p.m., sometimes thinking it is 60 degrees instead of 90 degrees. Remember that each hour mark is 30 degrees apart, and three such marks give 90 degrees. Using 180 degrees = π radians as the key conversion helps avoid mistakes.

Final Answer:
The circular measure of the angle between the hour hand and the minute hand at exactly 3:00 p.m. is π/2 radians.