At what times between 4 and 5 o’clock are the hands of a clock at right angles?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 4:05\(\frac{5}{11}\) and 4:38\(\frac{2}{11}\)
The minute hand gains 5.5° a minute on the hour hand (6° against 0.5°). Between 4 and 5 the hands are at right angles twice: once when the minute hand is 90° behind the hour hand and once when it is 90° ahead.
At 4:00 the hour hand is at \(4\times30^\circ=120^\circ\) and the minute hand at 0°: the minute hand is 120° behind
90° behind: it must gain \(120^\circ-90^\circ=30^\circ\), which takes \(\displaystyle\frac{30}{5.5}=\frac{60}{11}=5\frac{5}{11}\) minutes
90° ahead: it must gain \(120^\circ+90^\circ=210^\circ\), which takes \(\displaystyle\frac{210}{5.5}=\frac{420}{11}=38\frac{2}{11}\) minutes
So the times are 4:05\(\frac{5}{11}\) and 4:38\(\frac{2}{11}\)
