Problem of the Fortnight #4

The hour hand on a clock is three inches long and the minute hand is five inches. The two hands and the line segment connecting the tip of the two hands form a triangle. At what time between 12:00 and 12:30 does the triangle have the largest area?

Solution by Liu Xingcheng

Because the lengths of the hour and minute hands are fixed, say A and B, then the area of the triangle will be S = A * B * Sin<A, B>.

In order to get the maximum area, we need to know the maximum of Sin<A, B>,

Because the maximum of Sin X is 1, when the angle is 90 degrees,

Therefore, the problem is converted to: find the time when the angle between the hour hand and the minute hand is 90 degrees.

As known, the angular velocity of the hour hand is 0.5°per minute, and the angular velocity of the minute hand is 6° per minute, we need to find the time X such that:

6X - 0.5X = 90

X = 180/11 ≈ 16.36 (minutes).

The maximum area occurs at 12:16.36

Please check the answer with your watch.