How Close the Ships
Ship B is 100 miles east of ship A at noon. If ship B sails west at 10 mph and ship A sails south at 20 mph, at what time will the ships be closest to each other? At that time, what is the distance between the ships?
posted by Zaux at
6:38 PM
9 Comments:
2 hours.
Soz. 2 hours and 40√5 ≈ 89.44 miles.
Let the origin be at A when t = 0.
Using i to represent √(-1) and complex numbers a and b to represent the positions of A and B =>
a = -20it, b = 100 - 10t
where t is the time.
Let c be the distance between A and B,
c² = │a² + b²│ = 20² + (100 - 10t)²
c² = 500(t² - 4t + 20)
This is a maximum when dc/dt = 0
2 dc/dt = 1000t -2000 => t = 2 hours
c² = 500(4 - 8 + 20) = 800
c = 40√5
Can avoid calculus. Parabolas are symmetrical. The roots of t² - 4t + 20 are 2 +/- 4i, so the mid-point is at t = 2 as before.
I should have written 2c dc/dt, but it doesn't affect the solution, it was only a typo, and wouldn't affect the result anyway.
I have no idea why I chose to do it with complex numbers. The extra horse-power they provide wasn't needed.
Should have done it as: a = -20t, b = 100-10t.
c² = a² + b². The rest is identical.
right you are ... 2 hours and 40√5 miles
Sorry Zaux, have to disagree with the answer on a technicality....
2 hrs and 25.9070784 leagues (nautical), as I am sure you are aware, old sea dogs don't work in miles on the sea.....
Hi Karl ...
not being and old sea-dog, I will defer to your wisdom
noon ship travling south is mouving faster thab ship b noon 100 mi apart
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