At noon, ship A is 20 nautical miles due west of ship B.

Question

At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 15 knots and ship B is sailing north at 18 knots. How fast (in knots) is the distance between the ships changing at 5 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Answer 1

Let x be the distance along the east-west axis.

Let y be the distance along the north-south axis

Let s be the distance between A and B

dx/dt = -15 (negative since going west)

dy/dt = 18

Five hours later

x = -15(5)-20 = -95

y = 18(5) = 90

s² = x²+y²

Take the implicit derivative

2ss' = 2xx' + 2yy'

s' = (xx' +yy')/s

At 5:00:

s = √(95²+90²) = √(9025 +8100) ≈ 130.86 nautical miles

ds/dt = (-95(-15) + 90(18))/130.86 ≈ 23.27 knots