What is the measure, in degrees, of the acute angle formed by the minute hand and the hour hand on a standard clock when it indicates $9$:$40$?
I know that the hour hand moves $30$ degrees every hour, and the minute hand moves $360$ degrees every hour. So, at $9$:$00$, the hour hand is at $270$ degrees and the minute hand is at $0$ degrees, measuring from the positive x-axis. Then, at $9$:$40$, the hour hand has moved $\frac{40}{60} \times 30 = 20$ degrees, so it is at $270 + 20 = 290$ degrees. Meanwhile, the minute hand has moved $\frac{40}{60} \times 360 = 240$ degrees, so it is at $0 + 240 = 240$ degrees. The acute angle between the two hands is the difference between their angles, which is $290 - 240 = 50$ degrees. 