The circular clock above shows a time of exactly 3:30. What is the value of x?
Show Hint
A useful formula for the angle between clock hands is \( \text{Angle} = |30H - 5.5M| \), where H is the hour (1-12) and M is the minutes (0-59). For 3:30, \(H=3\) and \(M=30\). So, \( \text{Angle} = |30(3) - 5.5(30)| = |90 - 165| = |-75| = 75^\circ \).
Step 1: Understanding the Concept:
This problem asks for the angle between the hour and minute hands of a clock at 3:30. A clock face is a full circle of 360 degrees. Step 2: Key Formula or Approach:
We can calculate the position of each hand in degrees, measured clockwise from the 12 o'clock position.
- A clock face is divided into 12 hours, so the angle between each hour mark is \(\frac{360^\circ}{12} = 30^\circ\).
- The minute hand completes 360° in 60 minutes, so it moves at a rate of \(\frac{360^\circ}{60} = 6^\circ\) per minute.
- The hour hand completes 360° in 12 hours (720 minutes), so it moves at a rate of \(\frac{360^\circ}{720} = 0.5^\circ\) per minute. Step 3: Detailed Explanation: Position of the Minute Hand at 3:30:
At 30 minutes past the hour, the minute hand points directly at the 6. Its angle from the 12 is:
\[ 30 \text{ minutes} \times 6^\circ/\text{minute} = 180^\circ \]
Position of the Hour Hand at 3:30:
At 3:30, the hour hand has moved past the 3 and is halfway to the 4. The time is 3.5 hours past 12.
\[ 3.5 \text{ hours} \times 30^\circ/\text{hour} = 105^\circ \]
Alternatively, using minutes: The total number of minutes past 12 is \(3 \times 60 + 30 = 210\) minutes.
\[ 210 \text{ minutes} \times 0.5^\circ/\text{minute} = 105^\circ \]
Angle Between the Hands:
The angle \(x\) is the absolute difference between the positions of the two hands.
\[ x = | \text{Minute Hand Angle} - \text{Hour Hand Angle} | \]
\[ x = | 180^\circ - 105^\circ | = 75^\circ \]
Step 4: Final Answer:
The value of x, the angle between the hands at 3:30, is 75 degrees.
