Reference no: EM132356873

Assignment -

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 pads per million) at a point P(x, y) on the surface of seawater is approximated by

C (x, y) = e^{-(x^2+4y^2)}/10⁴

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

1. Sketch the surface described by C (x, y).

2. Sketch the level curves of the concentration function.

3. Identify the level curves of the concentration function. In other words, what are the equations for the level curves? (There should be a family of equations.)

4. What path would the shark take through the level curves that it intersects? Sketch an example path.

5. The gradient of the concentration points in the direction of the most rapid increase in concentration. What is the gradient of the concentration function?

6. If r(t) = x(t)i + y(t)j is a parameterization of the most rapid increase curve, then r'(t) is also tangent to curve and r'(t) = λ∇C. Use this information to solve for dy/dx.

7. Suppose a shark is at the point (x₀, y₀) when it first detects the presence of blood in the water. Find an equation of the shark's path, the most rapid increase curve, y = f(x), by setting up and solving a differential equation.

8. Suppose a shark is at the point (5, -4) when it first detects the presence of blood in the water.

a. What is the equation of the shark's path?

b. Sketch the equation of the shark's path along with the level curves of the concentration function.