Find the measure of the smaller angle formed by the hands of a clock at the following times. 3:15

Verified step by step guidance

1. Step 1: Understand that the clock is a circle, which is 360 degrees, and it is divided into 12 hours. Therefore, each hour represents an angle of \( \frac{360}{12} = 30 \) degrees.
2. Step 2: Calculate the angle of the hour hand at 3:15. At 3:00, the hour hand is at 90 degrees (3 hours \( \times \) 30 degrees per hour). At 3:15, the hour hand has moved an additional \( \frac{1}{4} \) of the way to the next hour, which is \( \frac{1}{4} \times 30 = 7.5 \) degrees. So, the hour hand is at \( 90 + 7.5 = 97.5 \) degrees.
3. Step 3: Calculate the angle of the minute hand at 3:15. The minute hand moves 360 degrees in 60 minutes, so it moves \( \frac{360}{60} = 6 \) degrees per minute. At 15 minutes, the minute hand is at \( 15 \times 6 = 90 \) degrees.
4. Step 4: Find the difference between the angles of the hour and minute hands. The angle between them is \( |97.5 - 90| = 7.5 \) degrees.
5. Step 5: Since 7.5 degrees is less than 180 degrees, it is the smaller angle formed by the clock hands at 3:15.

Key Concepts

Clock Angle Formula

The clock angle formula calculates the angle between the hour and minute hands of a clock. It is given by the formula: Angle = |(30*hour - (11/2)*minutes)|. This formula accounts for the fact that the hour hand moves as the minutes pass, providing an accurate measure of the angle formed at any given time.

Understanding Clock Positions

To effectively use the clock angle formula, one must understand the positions of the hour and minute hands. The hour hand moves 30 degrees for each hour (360 degrees/12 hours), while the minute hand moves 6 degrees for each minute (360 degrees/60 minutes). Knowing these movements helps in visualizing and calculating the angles accurately.

Smaller Angle Determination

When calculating the angle between the clock hands, it is essential to determine the smaller angle formed. Since a full circle is 360 degrees, if the calculated angle exceeds 180 degrees, the smaller angle is found by subtracting it from 360 degrees. This ensures that the answer reflects the acute angle between the hands.