At noon, ship A is 180 km west of ship B. Ship A is sailing south at 20 km/h and ship B is sailing north at 40 km/h. How fast is the distance between the ships changing at 4:00 PM? (Related Rates Question)

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anonymous · Answer

This question calls for a diagram, unfortunately I can't draw it on here.  If you drew a line segment representing the distance between ship A and ship B at the beginning, then you would write 180 km as that distance and put ship B going north at 40km/h and ship A going south at 20km/h.  To find the distance between them, use the pythagorean theorem.  180^2 + (40t+20t)^2 = c^2.  40t and 20t represent the rate times time for both ships.  The distance between the two ships is represented by...

$$h(t)=\sqrt{32400+3600t^2}$$

Find $$h'(t)=1/(2\sqrt{32400+3600t^2})*2*3600t=3600t/\sqrt{32400+3600t^2}$$

Plug t = 4 into the derivative h'(t) to find how fast the distance is changing between them at 4 pm.  It's 48 km/h.