The sum of the angles swept by minute and hour hands in 54 minutes is324°327°347°351°

1. Break Down the Problem

To determine the sum of the angles swept by the minute and hour hands in 54 minutes, we must calculate the angle for each hand separately and then combine these values.

2. Relevant Concepts

• The minute hand moves at a rate of $6^\circ$ per minute (since it completes $360^\circ$ in 60 minutes).
• The hour hand moves at a rate of $0.5^\circ$ per minute (since it completes $360^\circ$ in 12 hours, or $720$ minutes).

3. Analysis and Detail

Step 1: Calculate the angle swept by the minute hand

$\text{Angle by minute hand} = 6^\circ \times \text{minutes}$ Substituting $54$ minutes: $\text{Angle by minute hand} = 6^\circ \times 54 = 324^\circ$

Step 2: Calculate the angle swept by the hour hand

The hour hand moves $0.5^\circ$ for every minute. For 54 minutes: $\text{Angle by hour hand} = 0.5^\circ \times 54 = 27^\circ$

Step 3: Add the angles swept by both hands

$\text{Total angle} = \text{Angle by minute hand} + \text{Angle by hour hand}$ $\text{Total angle} = 324^\circ + 27^\circ = 351^\circ$

4. Verify and Summarize

The calculations for both angles have been verified, and their sum correctly represents the total angle swept by the hour and minute hands in 54 minutes.

Final Answer

The sum of the angles swept by the minute and hour hands in 54 minutes is $351^\circ$.