A clock is started at exactly 12:00 noon. By 5:10 p.m. on the same day, through how many degrees has the hour hand turned from its initial position at 12 o'clock?

Introduction / Context:
This question examines your understanding of how far the hour hand of a clock moves over a given time interval. Unlike the minute hand, which makes a full revolution every 60 minutes, the hour hand moves more slowly and covers 360 degrees in 12 hours. Knowing the rate of movement of the hour hand in degrees per minute allows you to convert elapsed time into angular displacement quickly and accurately.

Given Data / Assumptions:

• The clock starts at 12:00 noon, with the hour hand at 0 degrees reference position.

• We are interested in the position of the hour hand at 5:10 p.m. on the same day.

• The hour hand completes 360 degrees in 12 hours.

• Therefore, it moves 30 degrees per hour, or 0.5 degrees per minute.

Concept / Approach:
The idea is straightforward: find the total elapsed time in minutes from 12:00 noon to 5:10 p.m. and then multiply by the hour hand's rate of 0.5 degrees per minute. The product gives the total angular displacement of the hour hand from its starting position at 12:00. This avoids confusion with the minute hand and keeps the calculation simple and accurate.

Step-by-Step Solution:
Step 1: From 12:00 noon to 5:00 p.m. is 5 hours. Step 2: From 5:00 p.m. to 5:10 p.m. is an additional 10 minutes. Step 3: Total elapsed time = 5 hours 10 minutes = 5 * 60 + 10 = 310 minutes. Step 4: The hour hand moves 0.5 degrees per minute. Step 5: Total angle turned = 310 * 0.5 degrees. Step 6: 310 * 0.5 = 155 degrees. Step 7: Hence, by 5:10 p.m., the hour hand has rotated 155 degrees from its original 12 o'clock position.

Verification / Alternative check:
We can cross-check by breaking the movement into hours and minutes. In 5 hours, the hour hand moves 5 * 30 = 150 degrees. In the extra 10 minutes, it moves 10 * 0.5 = 5 degrees. Total = 150 + 5 = 155 degrees. This matches our earlier computation, confirming that 155 degrees is correct.

Why Other Options Are Wrong:
• 145 degrees: This would correspond to less elapsed time or a wrong rate per minute.
• 165 degrees: This overestimates the movement and would require 330 minutes (5 hours 30 minutes) at 0.5 degrees per minute, not the given 310 minutes.
• 175 degrees: This is even farther from the correct value and does not correspond to the actual elapsed time from 12:00 noon to 5:10 p.m.

Common Pitfalls:
Students sometimes mistakenly use the minute hand's rate (6 degrees per minute) instead of the hour hand's rate, leading to very large and incorrect angles. Another error is miscounting the total minutes between the given times, especially when converting hours to minutes. Carefully converting the time interval into minutes and using the correct rate of 0.5 degrees per minute ensures a correct answer.

Final Answer:
By 5:10 p.m., the hour hand of the clock has turned through 155 degrees from its starting position at 12:00 noon.