# calculus

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At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing
west at 25 knots and ship B is sailing north at 16 knots.
How fast (in knots) is the distance between the ships changing at 7 PM? (Note: 1 knot is a
speed of 1 nautical mile per hour.)

• calculus -

This and the other seem to be similar. Best to talk about the general method.

1. Find an expression for the distance between the two at time t.

In this case, with Pythagoras' help, that's

sqrt( (25t+30)^2 + (16t)^2)
= sqrt(881t^2 + 1500t + 900)
Nasty-looking thing. Call it
u = 881t^2 + 1500t + 900
and differentiate it.

du/dt = 1762t+1500
dy/du = 1/2sqrt(881t^2 + 1500t + 900)

so the whole thing is
(1762t+1500)/2sqrt(881t^2 + 1500t + 900)

Plug in t=7 and you're there.
=

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