5. (20 points) The clock shows the time as four hours and fifteen minutes. Determine the angle between the minute and hour hands at this moment.
Answer: $37.5^{\circ}$

Solution. Five minutes is $\frac{5}{60}$ of the circumference, which is $30^{\circ}$. The minute hand shows fifteen minutes, which is $90^{\circ}$. The hour hand has moved a quarter of the distance between four $\left(120^{\circ}\right)$ and five $\left(150^{\circ}\right)$ hours in fifteen minutes, so the hour hand shows $127.5^{\circ}$. The angle between the minute and hour hands at this moment: $127.5^{\circ}-90^{\circ}=37.5^{\circ}$.