# Thread: I need help with a distance type problem using knots and nautical miles

1. ## I need help with a distance type problem using knots and nautical miles

At 1300 hours a merchant ship sailing south at 18 knots is 40 nautical miles due east of a patrol boat travelling east at 24 knots. When will they be closest to each other?

Thank you for any help.

2. ## Re: I need help with a distance type problem using knots and nautical miles

$t=0$ is 1300

let the patrol boat's initial position be the origin, (0,0)

patrol boat's position at any time $t$ is $(24t,0)$

merchant ship's initial position is (40,0)

merchant ship's position at any time $t$ is $(40,-18t)$

distance between the two at any time $t$ ...

$d = \sqrt{(24t-40)^2 + [0-(-18t)]^2} = \sqrt{900t^2-1920t+1600}$

minimum distance will occur at the minimum value of the parabola, $900t^2-1920t+1600$, located at its vertex

when $t = \dfrac{-b}{2a} = \dfrac{1920}{2 \cdot 900} = \dfrac{16}{15} \, hrs = 1 \, hr \, 4 \, min$

time of CPA is 1404

Thanks man