1. 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. Suppose that in a certain case, the concentration of blood at a point P(x; y) on the surface of the seawater is given by, f(x; y) = 10^8 - 20x^2 - 40y^2 where x and y are measured in meters in a rectangular coordinate system with the blood source at the origin. Suppose a shark is at the point (100;500) on the surface of the water when it first detects the presence of blood. Find the equation of the shark's path towards the blood source. [Hint: You can use the fact that if y=g(x) satisfies g'(x) =(a/x)*g(x), then g(x) =Cx^a for some constant C. 2. So far in this class we have learned about limits and continuity of multi-variable functions, tangents planes, linear approximations, gradient vectors, directional vectors, and partial derivatives.[/b] 3. I know that the gradient vector points toward where the function is increasing most rapidly, to obtain the gradient vector I differentiated the equation f(x; y) for x to obtain fx = -40x and for y to obtain fy = -80y. Next I subbed in the sharks initial position (100;500) and found the gradient vector to be ∇f (100;500) = (-4000; -40000). The thing is my question asks for the equation for the sharks path to the blood source. I was thinking and wouldn't that mean its path is a straight line? If anyone has some guidance or knows what the next step is that would be much appreciated! Thanks.