Ship A is travelling south at the rate of 2 km/hr, at the instant that ship B, which is 32 miles south of ship A, is travelling east at rate of 4 km/hr.
a) Are they separating or approaching at the end of 2 hrs, and at what rate? b) At what time are they nearest together?
c) What is their minimum distance apart?
Initially distance between the ships is 32 mile = 32*1.6 km = 51.2 km
B is the initial position of ship-B
C is the position of ship-B after t hours
A is the position of ship-A after t hours
So AB = 51.2 - 2t and BC = 4t
So AC =
=> AC =
AC is the distance between the ships.
Let S = AC
As S is positive, S and S² have same type of monotonicity. So S and S² both will have same critical points.
Now,
S² =
Differentiating with respect to t
=>
a)
Since is negative at t = 2 , after 2 hours the ships will be approaching..
Now,
=>
=>
=>
=>
So the rate of approaching is 2.61 km/h
b)
For extremum value of S ,
=> 40t - 204.8 = 0 => t =
Now
So S² is minimum at t = 5.12 and hence S is also minimum at t = 5.12.
So distance will be minimum after 5.12 hours.
c)
When t = 5.12, S =
=
=
= 45.79
So the minimum distance is 45.79 km
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