Question

Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 4 km from the nearest point B on the shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D are 11 km apart. (Round your answers to two decimal places.)

Let W and L denote the energy (in joules) per kilometer flown over water and land, respectively. Assuming the bird's energy expenditure is minimized, determine a function for the ratio W/L in terms of x, the distance from B to C. W L =

Answer 1

The student's question requires the development of a function to represent the ratio of energy expended by a bird flying over water (W) to that over land (L) as it seeks the optimal path to conserve energy. This involves the use of Pythagoras' theorem in a right-angled triangle formed by the bird's flight over water and its path along the shoreline.

Step-by-step explanation:

The student's question involves applying principles of optimization to a scenario where a bird minimizes its energy expenditure. The bird avoids flying over water due to the increased energy cost associated with it and prefers flying over land. To solve for the function that describes the ratio of energy per kilometer flown over water (W) to that over land (L), we need to set up an equation that includes the distances flown over water versus land and the respective energy costs.

To determine the ratio W/L, we will regard the energy expended as a function of distance flown over water (x) and over land. Given that the bird is released from an island that is 4 km away from the nearest point B on the shore, and that it eventually arrives at its nesting area D, which is 11 km away from point B, we have a right-angled triangle. By minimizing the overall distance (and therefore energy), we can find the optimal path for the bird which will give us the ratio function.

To solve for W/L, we use Pythagoras' theorem for the water distance from the island to point C (x) measured along the shoreline. The land distance is then 11 - x km. Using a simple optimization problem setup, the calculation will lead to the energy ratio function dependent on x.