calculus question

Question

At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing west at 18 knots and ship B is sailing north at 22 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.)

Answer 1

Let x be ship A's distance and y be ship B's distance.

dx/dt = 18

dy/dt = 22

Let s be the distance between A and B.

Find: ds/dt

At 7:00 PM,

x = 30 + 18(7) = 66

y = 22(7) = 154

s²=x²+y²

s = √(66²+154²) = 167.547 NM

Take the derivative of both side of the equal sign.

2s ds/dt = 2xdx/dt +2ydy/dt

ds/dt = (xdx/dt +ydy/dt)/s

at 7:00 PM

ds/dt = (66(18)+154(22))/167.547≈27.3 Knots