A clock face has a 12 inch diameter, a 5.5-inch second hand, a 5 inch minute hand and a 3 inch hour hand. When it is exactly 3:30, calculate the rate at which the distance between the tip of any one of these hands and the 9 o'clock position is changing.
First consider the distance between the tip of the second hand and the 9 o'clock position; let us call this . Further, let be the angle between the 9 o'clock position and the second hand, and let be the time, in seconds. Then we have . If we use the Law of Cosines, we can get a formula for in terms of . That is, . (Draw a picture to help visualize this if you need to. The two smaller legs of the triangle will always be 6 and 5.5 inches, for the 9 o'clock position and second-hand tip, respectively. As changes, so does .) Using implicit differentiation, we get . Thus, the desired related rate is calculated using the chain rule: . At 3:30, we have and . Then .
The desired rate for the minute and hour hands is calculated in a nearly identical fashion. For the minute hand, we have and , so . At 3:30, and , so , or .
For the hour hand, we get and . Then . At 3:30, if we assume that , we see that this rate is 0. This is rather intuitional, as we may think of this as the peak in midair when we throw a ball up into the air; at this peak, the ball's velocity is 0. We can get a bit fancier though, as at 3:30 we could assume that the hour hand is halfway between the 3 and 4 o'clock positions. That is, at 3:30, , so . Then, , or , or .