Exercise 17

Let $z = f(x,y) = 50 - (x+1)^2 - (y+3)^2$ and $z = h(x,y) = 24 - 2x - 6y$.

Suppose a person is walking on the surface $z = f(x,y)$ in such a way that she walks the curve which is the intersection of $f$ and $h$.

1. Show that $x(t) = 4 \cos(t)$ and $y(t) = 4 \sin(t)$ is a parameterization of the “shadow” in the $xy$-plane of the curve that is the intersection of the graphs of $f$ and $h$.

2. Use the parameterization from part (a) to find the instantaneous rate at which her height is changing with respect to time at the instant $t = 2\pi/3$.