Question

At what time between 2 pm and 3 pm, will the hour and minute hands of a clock be in opposite directions (diametrically opposite)?

• A. 2:45 pm
• B. 2:44 pm
• C. 2:43 \(\frac{9}{11}\) pm
• D. 2:43 \(\frac{7}{11}\) pm

Hint:

For clock problems, use the formula for the angles between the hour and minute hands, and solve for the time when the angle is 180 degrees.

Answer 1

The Correct Option is D

Step 1: Understand the clock mechanics. The hands of the clock are diametrically opposite when the angle between them is 180 degrees. The hour hand moves 0.5 degrees per minute (since \( \frac{360^\circ}{12 \times 60} = 0.5^\circ \)) and the minute hand moves 6 degrees per minute (since \( \frac{360^\circ}{60} = 6^\circ \)). At 2:00 pm, the hour hand is at 60 degrees (since \( \frac{360^\circ}{12} \times 2 = 60^\circ \)). 
Step 2: Set up the equation for diametrically opposite hands. Let \( x \) be the number of minutes after 2:00 pm when the hour and minute hands are diametrically opposite (180 degrees apart). The positions of the hour and minute hands at time \( x \) are:
The position of the minute hand is \( 6x \) degrees.
The position of the hour hand is \( 60 + 0.5x \) degrees.
The angle between the hands at time \( x \) is given by: \[ \text{Angle between hands} = \left| 60 + 0.5x - 6x \right| = 180^\circ. \] Simplifying: \[ \left| 60 - 5.5x \right| = 180. \] This results in two equations: \[ 60 - 5.5x = 180 \quad \text{or} \quad 60 - 5.5x = -180. \] Solving both equations: For the first equation: \[ 60 - 5.5x = 180 \quad \Rightarrow \quad -5.5x = 120 \quad \Rightarrow \quad x = \frac{-120}{-5.5} = \frac{240}{11} \approx 21.82 \, \text{minutes}. \] For the second equation: \[ 60 - 5.5x = -180 \quad \Rightarrow \quad -5.5x = -240 \quad \Rightarrow \quad x = \frac{240}{5.5} = \frac{480}{11} \approx 43.64 \, \text{minutes}. \] Step 3: Conclusion. The correct time is \( x \approx 43.64 \) minutes after 2:00 pm, which is approximately: \[ 2:43 \, \frac{7}{11} \, \text{pm}. \] Thus, the correct answer is: \[ \boxed{2:43 \, \frac{7}{11} \, \text{pm}}. \]