Answer to Question #111387 in Algebra for julliet

Question

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?

Answer 1

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,

$\dfrac{B-b}{A-a}=\dfrac{-b}{A-x}$

$-b(x-a)=B(A-x)$

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

$y-b=\dfrac{-b}{x-A}(x-A)$

Now substituting the values,

$y-b=-\dfrac{B-b}{A-a}(x-A)$

$x=\dfrac{A-a}{b-B}(y-b)+A$

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

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

for maxima and minima

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

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

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