[RESOLVED] Help with math algorithm

[nuzzle]

the minimum will be at y = 8*x/63.

It depends on what you're optimizing. I've looked at minimizing w which is the distance between the runner and the approaching water when he starts running. For a given y and x this formula gives the w for which runner and water arrive at y simultaneously,

w = sqrt(x^2 + y^2) * 8 - y

The square root is the distance the runner runs to get to the riverbank a distance y downstream. Runner and water arrive at y at the same time but the water has travelled 8 times longer to get there hence the 8. Finally y is subtracted to give w, the distance between water and runner when he started to run.

In my view it makes most sense for the runner to aim for the y which gives the smallest possible w. W is the point of no return really. If the water gets closer you won't make it regardless of how you run. By running towards the y that puts w the closest to you will give you the best chance of making it regardless of how far away the water actually is when you start.

So what y gives the smallest w? Well according to my calculations it's y = x/sqrt(63). This differs from the y = 8*x/63 that D_Drmmr got so lets set x=1 and calculate w for the two y values.

w (1/sqrt(63)) = 7.937253933
w (8/63) = 7.937257808

Amazingly the results differ only after the fifth decimal. But mine is smaller so it's more optimal .