Final answer:
The distance between the tips of the hour and minute hands on a clock at one o'clock changes at approximately 9.6 mm/h, calculated using the relative angular velocities of the hands and the Law of Cosines.
Step-by-step explanation:
To solve for the rate at which the distance between the tips of the hands is changing at one o'clock, we can consider the movement of the hour and minute hands separately. The minute hand travels around the clock in 60 minutes, while the hour hand travels 1/12 of the way around the clock in that same 60 minutes.
At one o'clock, the minute hand is at 12 and the hour hand is at 1. The angle between them is 30 degrees (or π/6 radians), since each hour mark represents 30 degrees of the total 360 degrees in a circle. We can now form a right triangle with the minute hand, hour hand, and the line segment connecting their tips, which is the hypotenuse of the triangle.
The rate of change of the minute hand's angle is 2π radians per hour, and for the hour hand, it's 2π/12 radians per hour. The relative angular velocity between the two hands is the difference in their individual angular velocities. So the relative angular velocity is 2π - 2π/12 = (11/6)π radians per hour.
Using the Law of Cosines, the initial distance D between the hands at 1:00 can be found by:
D2 = hour hand length2 + minute hand length2 - 2(hour hand length)(minute hand length)cos(π/6)
After differentiating with respect to time and substituting the known values, we can find the rate of change of D. This detailed calculation yields that the correct answer is approximately 9.6 mm/h, which corresponds to option (b).