Last week, at Isaac's Turkey Farm, the turkeys were led on their final walk down Turkey Lane. Turkey Lane (see picture below) follows the path of a parabola ($y=x^2$) with the farm located at $\left ( 0,0 \right )$ and the slaughter house located at $\left ( 2,4 \right )$, where $x$ and $y$ are measured in miles. This year, one turkey decided to escape. At the point located 1 mile north and 1 mile east of the farm, Tom Turkey decided to branch off on his own. Being the last turkey in line, no one saw him make his escape. He moved along the tangent line (at the point where he escaped) to Turkey Lane in a NE direction.
- What is the equation of Tom's escape route? Explain how you arrived at this answer.
- At which point along his escape route was he closest to the slaughter house? Justify your answer using calculus.