You are building a new house on a cartesian plane whose units are measured in miles. Your house is to be located at the point $(2,0)$. Unfortunately, the existing gas line follows the curve $y= \sqrt{16x^2+5x+16}$. It costs 400 dollars per mile to install new pipe connecting your house to the existing line. What is the least amount of money you could pay to get hooked up to the system?
My try: find the derivative of the equation to find its critical points. And the compute the distance from the house to the critical point (local minimum).
However, I'm unable to compute the derivative. Kindly help with explanation.
Best Answer
You don't need to take the derivative of $\sqrt{16x^2+5x+15}$ to solve this problem.
If you hook up your house to the existing gas line at $(x,y) = (x,\sqrt{16x^2+5x+16})$, then the square of the length of the new pipe is $L^2 = (x-2)^2+(y-0)^2 = 17x^2+x+20$. At what value of $x$ is this minimized?
Once you've figured out the optimal value of $x$, you can compute the minimum length of the new pipe, and thus, the minimum cost to connect your house to the existing gas line.