Answer to Question #280921 in Calculus for Kelvin lee Boacon

Question

6. 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?

Answer 1

a)

initial distance between A and B is 32 km

distance between A and B after 2 hours:

$\sqrt{(32-2\cdot2)^2+(4\cdot2)^2}=29.12$ km < 32 km

so, A and B are approaching at the end of 2 hrs

rate of approaching at the end of 2 hrs:

$d'(2)=\frac{16\cdot2-2(32-2\cdot2)}{\sqrt{(32-2\cdot2)^2+(4\cdot2)^2}}=-0.82$ km/h

b)

distance between A and B as function of time:

$d(t)=\sqrt{(32-2t)^2+(4t)^2}$

$d'(t)=\frac{16t-2(32-2t)}{\sqrt{(32-2t)^2+(4t)^2}}=0$

$16t-2(32-2t)=0$

$t=64/20=3.2$ hours

c)

minimum distance apart:

$d(3.2)=\sqrt{(32-2\cdot3.2)^2+(4\cdot3.2)^2}=28.62$ km