recently got this question as a challenge from a tutorial. I used the cos rule and related rates to eventually get the formula dc/dt = dc/dtheta * dtheta/dt however I couldn't find dtheta/dt. How would you guys do this question?
Suppose you have a watch with a 15mm long minute hand and and a 12mm long hour hand. Furthermore, suppose the hour and minute hand move around the watch at a constant rate (so no ticking), and complete a full rotation in 12 hours and a one hour respectively. How fast is the distance between the end points of the hour and minute hand changing at 14:00?
The hour hand completes one rotation in $12$ hours. If we measure $t$ in minutes, that is $720$ minutes, over which time the angle increases by $2\pi$, so $\frac{d\theta}{dt}=\frac {2\pi}{720}$. You should use a different variable, like $\phi$ for the minute hand and compute $\frac{d\phi}{dt}$ the same way. Then write the distance as a function of $\theta, \phi$ and differentiate.