Clock: Rates of Change Problem

[Kilo]

Here's the problem:

The minute hand is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o'clock?

It's from my calculus textbook (Stewart)

I want to find how fast the distance changes relative to time, not the angle between the hands.

I decided to use the law of cosines and then take the derivative of it:
d=distance between hands, d'= rate of change of the distance, $$\theta$$=angle between hands, $$\theta$$'=rate of change of the angle

2dd'=2*4*8*sin($$\theta$$)$$\theta$$'

I can find d using law of cosines and angle $$\theta$$, but would it be okay to find $$\theta$$' by saying:

the hour hand travels $$\pi$$/6 per hour and the minute hand travels 2$$\pi$$ per hour...
$$\theta$$'= 2$$\pi$$-$$\pi$$/6=11$$\pi$$/6 per hour

I'm just not sure if that is how I should find $$\theta$$'

Thank you!

 

[HallsofIvy]

Yes, that argument is perfectly valid.