# It is known that homing pigeons tend to avoid flying over the water in the daytime, perhaps

because the downdrafts of air over water make flying difficult. Suppose a homing pigeon is
released on an island at point C , which is 3 km directly out in the water from a point B
onshore. Point B is 8 km downshore from the pigeon’s home loft point A. Assume that a
pigeon requires 1.28 times the rate of energy over land to fly over water. Toward what
point S downshore from A should the pigeon fly in order to minimize the total energy
required to get to home loft A ?

## To minimize the total energy required to get back to the home loft A, the homing pigeon should fly along the shoreline instead of directly over water.

Let's denote point S as the point where the pigeon should fly downshore from A to minimize energy. We need to find the position of point S.

Given:
- Point C is 3 km directly out in the water from point B.
- Point B is 8 km downshore from point A.
- The pigeon requires 1.28 times the rate of energy over land to fly over water.

To begin, let's consider the distance that the pigeon needs to fly over land and water to get to point S.

1. The distance from point A to point S along the shore is 8 km (since point B is 8 km downshore from point A).
2. The distance from point S to point C is the same as the distance from point B to point C since the pigeon should follow a straight line path along the shore.

Now, let's calculate the total energy required for the pigeon to fly over land and water, and compare it for two scenarios: flying directly across the water to point C and flying along the shore to point S.

Energy required to fly directly across the water to point C:
- Distance: 3 km
- Energy factor: 1.28 (since flying over water requires 1.28 times the energy of flying over land)
- Total energy required: 3 km * 1.28 = 3.84 km

Energy required to fly along the shore to point S:
- Distance: 8 km (along the shore from A to S) + 3 km (from S to C, which equals B to C)
- Energy factor: 1 (since the pigeon is flying over land)
- Total energy required: 8 km * 1 + 3 km * 1 = 8 km + 3 km = 11 km

Comparing the total energy required:
- Total energy required to fly directly across the water to point C: 3.84 km
- Total energy required to fly along the shore to point S: 11 km

Since the total energy required to fly along the shore to point S is less than the total energy required to fly directly across the water to point C, the pigeon should fly downshore towards point S to minimize the total energy required to get back to home loft A.

## To solve this problem, we need to determine the point S downshore from A where the pigeon should fly to minimize the total energy required to get back to the home loft.

We can start by analyzing the energy required for the pigeon to fly over land and water separately.

Let's consider the distance the pigeon needs to cover over land first. The distance from B to A is 8 km, so the pigeon will need to fly this distance over land.

Now, let's calculate the energy required to fly over water. The pigeon is currently at point C, which is 3 km out in the water from point B. Since the pigeon requires 1.28 times the rate of energy to fly over water compared to land, we can calculate the equivalent distance over land as 3 km * 1.28 = 3.84 km. Therefore, the pigeon needs to fly 3.84 km over land to cover the same amount of energy required to fly 3 km over water.

To minimize the total energy required, the pigeon should choose the path that minimizes the sum of these two distances. Let's call the distance from C to S as x km.

Therefore, the total energy required can be calculated as follows:

Energy over land = distance from B to A = 8 km
Energy over water = equivalent distance over land = 3.84 km

Total energy required = Energy over land + Energy over water = 8 km + 3.84 km

To minimize the total energy required, we should find the value of x that minimizes this sum. We can solve this by taking the derivative of the total energy with respect to x and setting it equal to zero. However, since this is a simple optimization problem, we can quickly see that the minimum value will be achieved when x is 4 km.

Therefore, the pigeon should fly to a point S that is 4 km downshore from point A to minimize the total energy required to get back to the home loft A.