A clock's minute hand has length 5 cm and its hour hand has length 4 cm. What is the distance between the....?

Starting t = 0 at 12:00 and measuring time in hours, the minute hand goes through 2pi every hour. So its angle (measured clockwise from 12:00) as a function of t is 2pi*t. This is pi/2 - theta where theta = the usual polar coordinate angle (measured counterclockwise from the x-axis).

So the position is (r cos(pi/2 - 2pi t), r sin(pi/2 - 2 pi t)) = (5 sin(2pi t), 5 cos(2 pi t) )

The hour hand goes through 2pi every 12 hours. Its clockwise angle as a function of t is 2pi t/12 = pi t/6. Its (x, y position) with the same reasoning is (4 sin(pi t/6), 4 cos(pi t/6) )

Now:

1. Use the distance formula to write down the distance between those two points as a function of t.

2. Take the derivative to answer the question. You want to know where the derivative is maximum.

Distance increases fastest when the hands are at

right angles (eg: 12/3)

at this point the distance between the tips is √(4²+5²)