Related Rates: The minute hand on a watch is 9 mm long and the hour hand is 6 mm long. How fast is the distance between the tips of the hands changing at one o'clock? (Round your answer to one decimal place.)

Answer 1

If the hands are separated by an angle θ, then using the law of cosines, the distance z between the tips is

z^2 = 6^2 + 9^2 - 2(6)(9) cosθ

The hour hand rotates at a speed of 2π/1440 radians/min, and the minute hand moves at 2π/60 radians/min.

So, figure θ at 1:00 and you know that dθ/dt = -23π/720

Now you can find z at 1:00 and you can find dz/dt using

2z dz/dt = 2(6)(9) sinθ dθ/dt