Answer to Question #111387 in Algebra for julliet

Question #111387
In a world with 2 villages A, B and a straight river, you have a lady in village A that need to fill water in the river and next go to village B. How will she find the point, x on the river such that the time walking in straight line from A to it Ax and from it to B Bx, will be the shortest, minimising Ax +Bx?
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Expert's answer
2020-04-23T15:17:32-0400

As per the question,

Let the co-ordinate of the village A is (A, b) and the co-ordinate of the village B is (a, B)

Let the co-ordinate of the point of on the river is O, (x, 0)

As per the question, point A and B are lying on the straight line, so slop of AO and OB will be same,

BbAa=bAx\dfrac{B-b}{A-a}=\dfrac{-b}{A-x}

b(xa)=B(Ax)-b(x-a)=B(A-x)

the equation of line passing from the point (x, 0) and and (A, b)

yb=bxA(xA)y-b=\dfrac{-b}{x-A}(x-A)

Now substituting the values,

yb=BbAa(xA)y-b=-\dfrac{B-b}{A-a}(x-A)

x=AabB(yb)+Ax=\dfrac{A-a}{b-B}(y-b)+A

Now, taking the differenciation of both side with respect to,

dx=AabBdydx=\dfrac{A-a}{b-B}dy

for maxima and minima

dxdy=0\dfrac{dx}{dy}=0

A=aA=a , so distance will be minimum, if a=0 and b=0

Hence, Ax+BxAx+Bx will be the minimum distance.


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