Question

Optimization

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=√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?

Answer 1

WELCOME TO OSS

Answer 2

you are optimising the distance between the pipe and the house at (2,0) subject to a constraint \(y = \sqrt{16x^2+5x+16}\), which is the path of the existing pipeline

the distance, s, from an arbitrary point (x,y) on that constraint curve to the house can be written \(s^2 = (x-2)^2 + y^2\)

replace x for y in the expression for \(s^2\) . do this by subbing in from \(y = \sqrt{16x^2+5x+16}\) or \(y^2 = 16x^2+5x+16\) , ie make it a single variable problem in x.

then solve for \(\dfrac {d(s^2)}{dx} = 0\)

NB we solve for \(s^2\). you can do it for \(s\), which might seem more intuitive, but the algebra is i think a wee bit lighter if you go with \(s^2\)

you can also keep it in 2 variables and use lagrange multipliers but not sure if that is "Calc 1" or not.