How fast (in knots) is the distance between the ships changing at 3 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Question

At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 25 knots and ship B is sailing north at 21 knots. How fast (in knots) is the distance between the ships changing at 3 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Answer 1

At 3:00 PM, A is 3(25) + 20 nMiles west from the origin of the x-y axis. B is 21(3) nMiles North. Let x = A and B = y. The distance between them is the hypotenuse of a right triangle.

x²+y² = s²

We need to find ds/dt at 3:00 PM.

Using implicit differentiation

2xdx/dt + 2ydy/dt = 2sds/dt --> ds/dt = (xdx/dt + ydy/dt)/s

s = √(95²+63²) ≈ 114

ds/dt = (95(25) + 63(21))/114 = 32.44 Knots