Question

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

• A. 4:05
• B. 4:38
• C. 4:55
• D. 4:22

Hint:

To quickly find clock angles without complex equations, remember that every 5-minute marker block on a clock face represents exactly $30^\circ$. At 4:00, the hands are 4 blocks ($120^\circ$) apart. To reach a $90^\circ$ separation, they need to be exactly 3 blocks apart!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Clock hands move at different, fixed speeds. The minute hand travels at $6^\circ$ per minute, while the hour hand travels at $0.5^\circ$ per minute. Therefore, the relative speed of the minute hand with respect to the hour hand is $5.5^\circ$ per minute. To be at a right angle ($90^\circ$) for the first time after 4 o'clock, the minute hand must reduce the initial angular distance and establish a exact $90^\circ$ separation behind or ahead of the hour hand.

Step 2: Key Formula or Approach:
The formula for the angle $\theta$ between the hands of a clock at $H$ hours and $M$ minutes is: $$\theta = \left| 30H - \frac{11}{2}M \right|$$ For the first right angle between 4 and 5 o'clock: Set $H = 4$ Set $\theta = 90^\circ$

Step 3: Detailed Explanation:
At exactly 4 o'clock, the hour hand is at the 4 mark, which corresponds to an angle of: \[ 4 \times 30^\circ = 120^\circ \] For the hands to form a $90^\circ$ angle for the first time, the minute hand must catch up enough to be exactly $90^\circ$ behind the hour hand. Thus, the net angular distance the minute hand needs to gain relative to the hour hand is: \[ 120^\circ - 90^\circ = 30^\circ \] Using our relative speed formula ($\frac{11}{2}^\circ$ per minute): \[ \text{Time } (M) = \frac{\text{Angular Distance to Gain}}{\text{Relative Speed}} \] \[ M = \frac{30}{\frac{11}{2}} = \frac{60}{11} \text{ minutes} \] Let us compute the exact value: \[ M = 5 \frac{5}{11} \text{ minutes} \approx 5.45 \text{ minutes} \] Looking closely at our time context, this means the first right angle occurs at approximately 4:05.

Step 4: Final Answer:
The hands of a clock will be at right angles for the first time at approximately 4:05.