Logistics problem?

Question

The Cactus Railway runs east and west. Dry Gulch is situated exactly 30 miles south of mile marker 70. Seco City is situated exactly 20 miles south of mile marker 190.
Where should a train station be built so that the sum of the distances to the two towns to the station is mimimum ?

Answer 1

i believe it is mile marker 125

Answer 2

If the station is built at mile marker x, the east-west distance to Dry Gulch will be x - 70 and the east-west distance to Seco City will be x - 190. The north-south distances are 30 and 20, respectively. Using the Pythagorean theorem, the total distance from the station to Dry Gulch is sqrt((x - 70)^2 + 30^2) and the distance to Seco City is sqrt((x - 190)^2 + 20^2). The sum of the distances is sqrt((x - 70)^2 + 30^2) + sqrt((x - 190)^2 + 20^2) = sqrt(x^2 - 140x + 4900 + 900) + sqrt(x^2 - 380x + 36100 + 400) = sqrt(x^2 - 140x + 5800) + sqrt(x^2 - 380x + 36500). The final step is to differentiate this expression with respect to x and solve for the derivative equal to zero, which indicates your extreme points; you'll need to check them to see which is the minimum.

f(x) = (x^2 - 140x + 5800)^0.5 + (x^2 - 380x + 36500)^0.5
f'(x) = 0.5*(x^2 - 140 + 5800)^(-0.5)*(2x - 140) + 0.5*(x^2 - 380x + 36500)^(-0.5)*(2x - 380) = (x - 70)/sqrt(x^2 - 140 + 5800) + (x - 190)/sqrt(x^2 - 380x + 36500)

If you set this equal to zero, get the fractions on opposite sides of the equation and square both sides, you should be able to cross-multiply to get a fourth order polynomial that you will need to solve.