Exercise 1


A person is swimming in the sea and wants to reach the coast as fast as possible.
The following coordinates are known:
Person(0,0) / Destination(-10,-10) / Coastline: y = -4
The speed at sea is 4km/h. The speed on land is 8km/h

Calculate the x-coordinate on the coastline.




We construct a function that represents the time to cover the distance:
* t(x): (distance from start position to coastline / speed at sea) + (distance from coastline to destination / speed at the beach) = 0
* start position = S(0,0) / place to reach land = K(x,-4) / destiny = D(-10,-10)
=> t(x): sqrt((x-0)²+(-4-0)²)/4 + sqrt((-10-x)²+(-10+4)²)/8 = 0
=> t(x): sqrt(x²+16)/4 + sqrt(136+x²+20x)/8 = 0
t'(x) = 0 => x = - (80·sqrt(2867)/27 - 4232/27)^(1/3) + (80·sqrt(2867)/27 + 4232/27)^(1/3) - 22/3
=> x = -1.766999492...