To show: The total energy used by the bird is .
The energy of the bird requires to fly over the land and to fly over the water.
All the distances related to the bird are given in the figure below,
Let x be distance between point B and C.
Use the Pythagoras theorem to find the length AC.
The length of the side BD is 12mi and the length of the side BC is x mi.
Then, the length of the side CD is,
The bird flies from the point C to the point D on the land with the energy .
To find the energy for , multiply by both side of the equation (2),
The bird flies from the point A to the point C on the water with the energy .
To find the energy for , multiply on both sides of the equation (3),
Then the total energy is,
Substitute for and for in the equation (4),
Thus, the total energy is .
To find: The path that minimizes the energy expenditure.
The bird flies till the point C on shoreline which is 5.013 miles from point B and then flies along the shoreline.
From the part (a), the total energy is .
To find the minimum energy sketch the graph of .
The function contains the variable x is the length of the side BC.
The local minimum value of the function is the least finite value where the value of the function at the any number is less than to the original function.
The condition for local minimum is,
Using online graphing calculator, sketch the graph of the function as shown in the figure below.
From the above figure, it can be observed that the least peak occurs at the point .
Then, the minimum energy expenditure is 168.99 at .
Thus, the point C is 5.103 mi away from point B.
Therefore, the bird flies till the point C on shoreline which is 5.013 miles from point B and then flies along the shoreline.
Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!