## Problem 5.

On the bank of a 100-meter wide river is a power plant, and on the opposite bank, 500 meters upstream, a factory is being built. Knowing that the river is straight between the power plant and the factory, that the cost of laying cables along the bank is $9 €$ per meter, and that the cost of laying cables over the water is $15 €$ per meter, what is the length of the most economical cable run between the power plant and the factory?
Solution:

Each route will have a journey consisting of a river segment, advancing a distance of b meters, and one or two segments along the bank covering the remaining 500 - b meters. The journey of such a route will be L(b) and the cost g(b).

![](https://cdn.mathpix.com/cropped/2024_04_26_736f86425b14d88c7083g-3.jpg?height=340&width=919&top_left_y=2217&top_left_x=654)

The length of the most economical possible route between the power plant and the factory is 550 meters.