The minute hand of a clock is twice as long as the hour hand. At what time, between 00:00 and when the hands are next aligned (just after 01:05), is the distance between the tips of the hands increasing at its greatest rate?

Question

The minute hand of a clock is twice as long as the hour hand.  At what time, between 00:00 and when the hands are next aligned (just after 01:05), is the distance between the tips of the hands increasing at its greatest rate?

anonymous · Answer

Hint: One approach is to express the distance between the tips of the hands in terms of the lengths of the hands and the angle between them.  Differentiate the distance with respect to the angle.  Use the law of sines (also known as the sine rule) to simplify the resulting expression.
Or, you may be able to find a non-calculus approach.