Homing pigeons avoid flying over large bodies of water, preferring to fly around them instead. Assume that a pigeon released from a boat
6
6
miles
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
12 miles from point A, the point on the shore closest to the boat, and if a pigeon needs
four thirds
4
3
as much energy per mile to fly over water as over land, find the location of point P, which minimizes energy used.
If we let x = the distance AP, then the energy used is
z = 4/3 √(x^2+6^2) + 12-x
dz/dx = 4x/(3√(x^2+36)) - 1
so find where dz/dx=0
Fix your post, typing in a normal way, and describe the diagram which we obviously cannot see
To find the location of point P that minimizes the energy used by the pigeon, we need to consider the total energy required for the pigeon to fly to point P and then along the straight edge of the lake to its home at L.
Let's break down the problem step by step:
1. First, we need to find the energy required for the pigeon to fly from the boat to point P on the shore. Since the pigeon avoids flying over the lake, it will try to fly the shortest distance over land.
2. The pigeon is released 6 miles from the shore, and we know that the distance between point A (the closest point on the shore) and point L (the home) is 12 miles. This means that the total distance the pigeon needs to fly over land is 12 - 6 = 6 miles.
3. Next, let's consider the energy required for the pigeon to fly along the straight edge of the lake from point P to its home at L. The distance between P and L is also 6 miles.
4. Given that the pigeon needs 4/3 as much energy per mile to fly over water as over land, we can calculate the total energy required to fly from point P to L over water as (4/3) * (6 miles) = 8 miles.
5. Now, we can compare the total energy required for the pigeon to fly from the boat to point P over land and then from point P to its home at L over water with the total energy required for the pigeon to fly the entire distance over land.
6. The total energy required for the pigeon to fly the entire distance over land is (6 miles) * (1 unit of energy per mile over land) = 6 miles.
7. To minimize the energy used, the pigeon would prefer to fly the shortest distance. Therefore, it should choose the path where the total energy required is the smallest.
Considering the total energy required for the pigeon to fly the entire distance over land is 6 miles, and the total energy required to fly from the boat to P over land (6 miles) plus from P to L over water (8 miles) is 14 miles, it is more energy efficient for the pigeon to fly directly to point L along the straight edge of the lake. This means that the location of point P, which minimizes energy usage, is the same as the location of point L.