A company wishes to run a utility cable from point A on

Question

the shore to an installation at
point B on the island. The island is 6 miles from the shore. It
please explain it for me step by step without link it to a similar question: costs $400 per mile to run the cable on land and $500 per mile
underwater. Assume that the cable starts at A and runs along
the shoreline, then angles and runs underwater to the island.
Find the point at which the line should begin to angle in order
to yield the minimum total cost. the length between A AND C is 7

Answer 1

I assume that C is the point on shore closest to B.

So, if the pipeline leaves the shore at point P, between A and C, and the distance PC = x, the length p of the pipeline is

p(x) = (7-x) + √(x^2+36)

now we see that the cost c of the pipeline is

c(x) = 400(7-x) + 500√(x^2+36)

Now just find the minimum of c(x).