Related Rates problem: What is the best way to solve this problem?

Question

At noon, ship A is 50 nautical miles due west of ship B. Ship A is sailing west at 24 knots and ship B is sailing north at 23 knots. How fast (in knots) is the distance between the ships changing at 3 PM?

The distance is changing at  ?  knots.

(Note: 1 knot is a speed of 1 nautical mile per hour.)

Answer 1

Draw a grid

let x be ship A's distance in the x direction

let y be ship B's distance in the y direction

let s be the distance between the two ships.

This forms a right triangle with s as the hypotenuse

x²+y²=s²

dx/dt = -24 knots

dy/dt = 23 knots

We need to find ds/dt.

Take the derivative of both sides of the equation

2x dx/dt +2y dy/dt = 2s ds/dt

ds/dt = (x dx/dt + y dy/dt)/s

After 3 hours,

x = -50 -24(3) = -122 nautical miles

y = 23(3) = 69 nautical miles

s = √(122²+69²)≈140.16 nautical miles

ds/dt = (-122(-24) + 69(23))/140.16 ≈32.2 knots