At 8 P.M. an oil tanker traveling west in the ocean at 17 kilometers per hour pa

Question

At 8 P.M. an oil tanker traveling west in the ocean at 17 kilometers per hour passes the same spot as a luxury liner that arrived at the same spot at 7 P.M. while traveling north at 23 kilometers per hour. If the "spot" is represented by the origin, find the location of the oil tanker and the location of the luxury liner t hours after 7 P.M. Then find the distance D between the oil tanker and the luxury liner at that time.

D(t) =

At what time were the ships closest together? (Hint: Minimize the distance (or the square of the distance!) between them.)

The time is :

Explanation / Answer

At 8 pm the oil tanker was at the spot therefore it was 17 km east of the spot at 7 pm

Now at 7 pm the luxury liner was at the spot travelling at 23 kmph North

Now distance of oil tanker after t hours from 7pm will be 17(t-1) (Answer) km west of the spot

And distance of luxury liner after t hours after 7 pm will be 23t (Answer) km North of the spot

Now distance between these points will be D^2 = (23t)^2 + [17(t-1)]^2 (From Pythagoras Theorem)

or D = [(23t)^2 + [17(t-1)]^2]^1/2 (Answer)

Now the shortest distance will be when dD/dt = 0

therefore dD/dt = [ 1058t + 578t - 578] = 0

therefore t = 0.3533 hours = 21.19 minutes

Hence T = 7:21 PM (Answer)

therefore the time is 7:21 when the oil tanker and the luxury liner are the closest

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