An oil pipeline will connect points A and B which are 3 km apart and on opposite banks of a 1 km wide river. From point A to point C on the opposite bank, the pipeline will run under the water. From point C to point B the pipeline will run above ground.

Question

The cost of running the pipeline under water is three times the cost of running it above ground.

Determine an equation that can be used to minimize the cost. You need not solve the problem.

Answer 1

Your language is ambiguous, but this kind of problem usually specifies that B is 3 km from a point O directly across the river from A.
If that is the case, then let OC=x and thus CB = 3-x
The cost function is then
f(x) = 3√(x^2+1) + (3-x)

If you meant to say that the line-of-sight distance AB=3, OB = √8, rather than 3.