Marine biologists have determined that when a shark detects the presence of bloo

Question

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 the most rabidly. Based on certain tests, the concentration of blood (in parts per million) at a point P(x, y) on the surface of the seawater is approximated by C(x, y) = e^-(x^2 + 2y^2)/10^4 where x and y are measure in meters with the blood source at the origin. a. Identify the level curves of the concentration function and sketch 5 curves. Also sketch a possible path that a shark will follow to the source, if the shark first detects the blood at the point (x_0, y_0). b. Find an equation of the shark's path in the xy plane that you drew for part a.

Explanation / Answer

Level curve for concentration will be of the form

x^2 + 2y^2 = C

So it is an ellipse with centre at origin

The shark swims in a direction where concentration increses most rapidly,

dc/dx = e^-(10^-4 (x^2+2y^2)) * (-2x) / 10^4

dc/dy = e^-(10^-4 (x^2+2y^2)) * (-4y) / 10^4

Concentration increases most rapidly when it is normal at point (x0 and y0)

So dy/dx = x/2y

So on a line which is perpendicular = -2y0/ x0

Passes through (x0 and y0)

So (y - y0) *x0 = (x-x0) * (-2y0) is the trajectory

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.