One ship is sailing south at a rate of 5 knots, an and another is sailing east at a rate of 10 knots. At 2 P.M. the second ship was at the place occupied by the first ship one hour before. At what time was the distance between the ships not changing?
If we draw a (traiangular) diagram as depicted in the ASCII art below:
y| / z
Knowing the ship speed (at any time) and its positions at 2PM, we can say
x = 10(t-2) and y = 5(t-1), (z = distance between the two ships)
where t is the time in hours after 1PM ( so that at t=2, x = 5, y = 0 and dx/dt=10, dy/dt=5).
The variables x, y and z obey the Pythagoras theorem (at all time t)
x^2 + y^2 = z^2
Implicitly differentiating the equation with respect to t, we get
2x dx/dt + 2y dy/dt = 2z dz/dt.
So asking when the distance between the shops not changing (albeit momentarily) is asking for the moment
when dz/dt = 0. Now if we substitute dz/dt=0 and all the other information we know we ended up with
2* 10(t-2)*10 + 2* 5(t-1)* 5 = 2 * z * (0)
=> 200 (t-2) + 50 (t-1) = 0
=> 4 (t-2) + (t-1) = 0 (after diving by 50 on the equation)
=> 4t - 8 + t - 1 = 0 (remove parentheses)
=> 5t = 9
=> t = 9/5
t=0 corresponds to the time 1:00PM, so t=9/5 = 1 hour and 48 minutes corresponds to the time 2:48PM.
Hence at exactly 2;48PM the distance between the ships was not changing.