Building a highway at the minimum cost

Question

I am asked the following question:

You are responsible for building a highway that connects A to B. There's an old highway 50 miles south that can be restored at the cost of \$300.000,00 per mile. Building a new path requires \$500.000,00 per mile. Find the optimal path that minimizes the cost.

So we can call the projections of A and B A' and B', and the midpoints between them X and Y.

Using a straight path from A to B would require \$75.000.000,00 and going AA'B'B would require \$95.000.000,00. Sonlet's find an optimal point on the middle.

The cost function is given by

$$ C = 2 \cdot 500000 \sqrt{50^2+x^2} + 300000(150-2x)\\ C' = \frac{10x}{\sqrt{x^2+2500}}-6=0 $$

If we say $C'=0$ we don't find any critical points. Did I do a miscalculation? And if not, what is the conclusion of the problem?

Thank you.

Answer 1

Your equation $C'=0$ does have a solution. Add $6$ to both sides, multiply by $\sqrt{x^2+2500}$, then square both sides, getting $$ 100x^2 = 36(x^2+2500) $$ which has the solution $x=300/8$.