How fast (in knots) is the distance between the ships changing at 3 PM?

Question

At noon, ship A is 50 nautical miles due west of ship B. Ship A is sailing west at 23 knots and ship B is sailing north at 18 knots. (Note: 1 knot is a speed of 1 nautical mile per hour.)

Answer 1

Let x be the distance between A and the initial location of B. Let y be the distance between B and its initial location. After 3 hours, x =50+3(23) =119 nautical miles, and y = 3(18)=54 nautical miles. The distance between the two ships is the hypotenuse of a right triangle = s.

dx/dt = 23

dy/dt = 18

s²=x²+y²

Taking the derivative with respect to t of both side of the equation.

2sds/dt = 2xdx/dt + 2ydy/dt

s = √(119²+54²)

ds/dt =(1/s)(x dx/dt + y dy/dt)

Just substitute in the values to get ds/dt which is what the problem is requesting.