### 44th Putnam 1983

**Problem A2**

A clock's minute hand has length 4 and its hour hand length 3. What is the distance between the tips at the moment when it is increasing most rapidly?

**Solution**

Answer: √7.

Let the angle between the hands be θ. Then the distance between the tips is √(25 - 24 cos θ). Differentiating, the rate of increase is 12 dθ/dt sin θ / √(25 - 24 cos θ). Differentiating again, this is a maximum when cos θ (25 - 24 cos θ) = 12 sin^{2}θ, and hence when 12 cos^{2}θ - 25 cos θ + 12 = 0, or (3 cos θ - 4)(4 cos θ - 3) = 0. We cannot have cos θ > 1, so the maximum is when cos θ = 3/4 and the distance is √(25 - 24.3/4) = √7.

44th Putnam 1983

© John Scholes

jscholes@kalva.demon.co.uk

16 Jan 2001