A North-South highway intersects an East-West highway at a point P. A vehicle crosses P at 1:00 p.m., traveling east at a constant speed of 60km/h. At the same instant, another vehicle is 5 km north of P, traveling south at 80 km/h. Find the time when the two vehicles are closest to each other and the distance between them at this time.
Answer 143.
Treating P as origin and direction north as +ve y-axis, direction east as +ve x-axis,
the position of both the vehicles after time t after 1 p.m. is
(60t, 0) and (0, 5 - 80t)
Square of the distance between the two at time t is
s^2 = (60t)^2 + (5 - 80t)^2
=>s^2 = (100t)^2 - 800t + 25
s is least when s^2 is least
For s^2 to be minimum,
d/dt(s^2) = 0 and d^2/dt^2 (s^2) >0
d/dt(s^2) = 0
=> 2*(100t)*100 - 800 = 0
=> t = 800/20000 = 1/25 hr
=> t = 60/25 min = 2.4 min = 2 min 24 sec
d^2/dt^2 (s^2) = 20000 > 0
=> the vehicles are closest at time 1:02:24 p.m.
The closest distance is given by
s^2 = (100*/25)^2 - 800/25 + 25 = 9
=> s = 3 km.
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