Question

At what time between 3:00 and 4:00 will the minute hand and hour hand of a clock be exactly opposite each other?

• A. 3:15 PM
• B. 3:20 PM
• C. 3:30 PM
• D. 3:49 PM

Hint:

Use the standard equation for hand angles: \( \big|30h - 5.5t\big| = \theta \) (degrees), where \(h\) is the hour and \(t\) minutes past \(h\). For opposite hands, set \(\theta=180^\circ\).

Answer 1

The Correct Option is D

Step 1 (Set up the angle equation).
At \(t\) minutes after \(3{:}00\), the hour hand is at \(30\times 3 + 0.5t = 90 + 0.5t^\circ\).
The minute hand is at \(6t^\circ\).
Opposite hands \(\) angle difference \(= 180^\circ\): \[ |\, (90 + 0.5t) - 6t \,| = 180. \] Step 2 (Solve for \(t\)).
\(|\,90 - 5.5t\,| = 180\).
Case 1: \(90 - 5.5t = 180 -5.5t = 90 t = -16.\overline{36}\) (reject; negative).
Case 2: \(5.5t - 90 = 180 5.5t = 270 t = \dfrac{270}{5.5} = \dfrac{540}{11} = 49 \dfrac{1}{11}\ \text{min}.\)
Step 3 (Write exact time).
\(49 \dfrac{1}{11}\) minutes \(= 49\) minutes \(+\ \dfrac{60}{11}\) seconds \(= 49\) minutes \(+\ 5.\overline{45}\) seconds.
So time \(\approx 3{:}49{:}05\ \text{PM}\).
Step 4 (Compare with options).
None of the choices matches \(3{:}49\frac{1}{11}\ \text{PM}\). The closest given option is \(3{:}45\ \text{PM}\), but it is not exact.
\[ {3{:}49\frac{1}{11}\ \text{PM (not among options)}} \]