Chapter 14.P, Problem 2P

Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the blood increases most rapidly. Based on certain tests, the concentration of blood (in parts per million) at a point $P\left(x,y\right)$ on the surface of seawater is approximated by

$C\left(x,y\right)=e^{-(x^2+2y^2)/{10}^4}$

where x and y are measured in meters in a rectangular coordinate system with the blood source at the origin.

(a) Identify the level curves of the concentration function and sketch several members of this family together with a path that a shark will follow to the source.

(b) Suppose a shark is at the point $\left(x_0,y_0\right)$ when it first detects the presence of blood in the water. Find an equation of the shark’s path by setting up and solving a differential equation.