Question

A bridge is 30ft above a canal. A boat is travelling at 10ft/sec passes under the centre of the bridge at the same moment that a man walking 5ft/sec reaches that point. How rapidly are they separating 3seconds later?

Answer 1

This is a Pythagoras problem. The man's direction of travel (perpendicular to the boat's) forms one side of a triangle, and the boat's the other, with the distance between them forming the hypotenuse.
So:
\[\Large \sqrt {{5^2} + {{10}^2}} = \sqrt {125} = 5\sqrt 5 \cong 11.18\,{\rm{ft/sec}}\]
Alternatively (or as a check), take their respective distances at 3 seconds (3*5 = 15, 3*10 = 30) applying the same reasoning, and then their distances 1 second later and determine how far they would have separated in that 1 second:
\[\Large \begin{array}{l}
\sqrt {{{15}^2} + {{30}^2}} = \sqrt {1125} \\
\sqrt {{{20}^2} + {{40}^2}} = \sqrt {{\rm{2000}}} \\
\frac{{\sqrt {2000} - \sqrt {1125} }}{1} = 5\sqrt 5 \cong 11.18\,{\rm{ft/sec}}
\end{array}\]