Project D03 - A Bird Over Troubled Water

A bird, located at the point C over the center of a lake, must eventually get to the point Q.

Because of the air currents over the lake, the bird will require more energy to fly straight to Q, than to fly straight to an intermediate point P and then along the arc of a circle to Q.

The bird will fly a distance l from the point C, and at this distance, the air currents over the lake will no longer affect its flight. Let x represent the first coordinate of the point P, and let el represent the energy required for the bird to fly one meter over land. Your goal will be to find the x-value that minimizes the amount of energy required by the bird.

  1. Suppose that the energy required to fly one meter over the water, ew, depends on x in such a way that ew(0) = 0 and ew(l) = 2el. Find a linear function that satisfies these conditions.
  2. Find a function of x that gives the length of the arc from P to Q.
  3. Find a function of E(x) that gives the total energy required for the bird to reach point Q.
  4. Suppose that el = 1 (don't worry about the units) and l = 1000 m. Sketch the graph of the function E for 0 £ x £ 1000.

  5. Find x so that the amount of energy is minimized.

  6. At what angle (from due north) should the bird fly to get to the point P.
  7. Write a paragraph discussing the model. How realistic is the model? What changes could be made to make it more realistic? Do birds actually try to minimize energy requirements? etc.