Question

Homing pigeons avoid flying over large bodies of water, preferring to fly around them instead. Assume Point P is miles away from point A. that a pigeon released from a boat 6 miles from the shore of a lake (point B in the figure) flies first to point P on the shore and then along the straight edge of the lake to reach its home at L. If L is 12 miles from point A, the point on the shore closest to the boat, and if a pigeon needs 4/3 as much energy per mile to fly over water as over land, find the location of point P, which minimizes energy used.

Answer 1

To minimize the energy used, we need to find the value of x that minimizes the expression E. By taking the derivative of E and setting it equal to 0, we can find the critical points. The location of point P that minimizes energy used is 2.5 miles from the shore.

Step-by-step explanation:

To find the location of point P that minimizes energy used, we need to find the shortest distance for the pigeon to fly over water. Let's assume that the pigeon flies over water for x miles and over land for (6 - x) miles. Since the pigeon needs 4/3 as much energy per mile to fly over water as over land, the total energy used can be expressed as:

E = (4/3)(x) + (6 - x)

To minimize the energy used, we need to find the value of x that minimizes the expression E. Taking the derivative of E with respect to x and setting it equal to 0, we can find the critical points. By solving the equation, we find that x = 2.5 miles. Therefore, the location of point P that minimizes energy used is 2.5 miles from the shore.