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.


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