Question

At what time between 4 and 5 o’clock will the hands of a clock be at right angles? (Approximately)

• A. \(4:05 \)
• B. \(4:38 \)
• C. \(4:55 \)
• D. \(4:45 \)

Hint:

You can use visual logic to eliminate wrong choices immediately! At 4 o'clock, the hour hand is at 4 and the minute hand is at 12. - At 4:05, they are very close together (acute angle). - At 4:45, the minute hand is at 9, which forms a broad obtuse angle. - At 4:38, the minute hand is near the 7.5 mark, which creates an exact perpendicular \(90^\circ\) configuration with the hour hand (which has moved past 4)!

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The hands of a clock move at different angular speeds. The minute hand moves at a rate of \(6^\circ\) per minute, while the hour hand travels at \(0.5^\circ\) per minute. This creates a relative separation speed of \(5.5^\circ\) per minute. A right angle configuration means the structural separation between the hour hand and minute hand is exactly \(90^\circ\) (equivalent to a space of 15 minute spaces).

Step 2: Key Formula or Approach:
To find the exact time location where a specific angle is created, use the standard clock relative velocity formula: \[ T = \frac{2}{11} \left(30H \pm \theta\right) \] where \(H\) is the starting hour base (which is \(4\) in this problem), and \(\theta\) is the targeted angle (which is \(90^\circ\)).

Step 3: Detailed Explanation:
Let's substitute our parameters into the formula. Since the hands create a right angle twice every hour, we evaluate the addition configuration (\(+\)) to look for the later time position that aligns with our options: \[ H = 4, \quad \theta = 90^\circ \] \[ T = \frac{2}{11} \left(30(4) + 90\right) \] \[ T = \frac{2}{11} \left(120 + 90\right) \] \[ T = \frac{2}{11} \times 210 \] \[ T = \frac{420}{11} \] Now divide \(420\) by \(11\) to extract the mixed fraction values: \[ T = 38 \frac{2}{11} \text{ minutes} \] This value simplifies approximately to \(38\) minutes past the hour. Therefore, the clock hands form a right angle at approximately 4:38, which corresponds to option (B).

Step 4: Final Answer:
The hands of the clock will be at right angles at approximately 4:38.