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 7 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.)
(a) In general, if it takes 1.3 times as much energy to fly over water as land, to what point C should the bird fly in order to minimize the total energy expended in returning to its nesting area?
(b) 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.
(c) What should the value of W/L be in order for the bird to fly directly to its nesting area D?
(d) If the ornithologists observe that birds of a certain species reach the shore at a point 6 km from B, how many times more energy does it take a bird to fly over water than land?
As mentioned, let BC = x
distance over water: √(x^2+49)
distance over land: 11-x
(a) energy expended:
y = 1.3√(x^2+49) + 11-x
dy/dx = 1.3x/√(x^2+49) - 1
y'=0 at x = 70/√69
what do you get for the other parts?
To answer these questions, we need to consider the energy expended by the bird while flying over water and land. Let's break down the problem step by step:
(a) To minimize the total energy expended, we need to find the optimal point C on the shoreline for the bird to fly to. The bird should fly over land until it reaches the point where the energy required to fly over water is equal to the energy required to fly over land.
Let B be the point on the shoreline nearest to the island, C be the optimal point on the shoreline, and D be the nesting area. We are given that B and D are 11 km apart and the island is 7 km away from B.
To find optimal point C, we can set up an equation based on the energy expended over water and land. Let L be the energy expenditure per kilometer flown over land, and W be the energy expenditure per kilometer flown over water.
The energy expended flying from the island to point C over land is 7 km * L. The energy expended flying from point C to the nesting area D along the shoreline is (11 - x) * W, where x represents the distance from B to C.
Since it takes 1.3 times more energy to fly over water than land, we can set up the equation:
7 L + 1.3 * (11 - x) L = x W
Simplifying this equation, we get:
7 L + 14.3 L - 1.3 x L = x W
21.3 L - 1.3 x L = x W
20 L = x W
W/L = 20/x
Therefore, the bird should fly to the point C on the shoreline that is 1/20th of the distance from B to D.
(b) By rearranging the equation W/L = 20/x, we can express W in terms of L and x:
W = 20L/x
So, the function for the ratio W/L in terms of x is W/L = 20/x.
(c) To find the value of W/L that allows the bird to fly directly to its nesting area D, we need to set x to the total distance from B to D, which is 11 km. Substituting x = 11 into the equation W/L = 20/x, we get:
W/L = 20/11
(d) If the birds of a certain species reach the shore at a point 6 km from B, we can substitute x = 6 into the equation W/L = 20/x to find the ratio of energy expended over water to land:
W/L = 20/6 = 10/3
So, it takes approximately 3.33 times more energy for the bird to fly over water than land.
(a) In order to minimize the total energy expended, the bird should fly to the point C that is equidistant from both B and D. This means that the bird should fly towards the midpoint between B and D.
The distance from B to D is 11 km, so the midpoint is at a distance of 11/2 = 5.5 km from either end.
Therefore, the bird should fly to a point C that is 5.5 km from point B on the shoreline.
(b) Let's assume that the distance from B to C is x km. The distance from C to D would then be (11 - x) km.
The total energy expended can be calculated as follows:
Energy over water = W * x km
Energy over land = L * (11 - x) km
The total energy expended is the sum of these two:
Total energy = W * x + L * (11 - x)
Since it takes 1.3 times as much energy to fly over water as land, we can write:
W = 1.3L
Substituting this into the equation for total energy, we get:
Total energy = (1.3L) * x + L * (11 - x)
Total energy = 1.3Lx + 11L - Lx
Total energy = 0.3Lx + 11L
Thus, the function for the ratio W/L in terms of x is:
W/L = 0.3x + 11
(c) If the bird were to fly directly to its nesting area D, then the distance from B to C would be equal to the distance from C to D. Let's call this distance d.
So, x = d and 11 - x = d.
Simplifying these equations, we get:
x = d
11 - x = d
Substituting these into the equation for the ratio W/L from part (b), we have:
W/L = 0.3d + 11
Since the bird would not be flying over water in this case, the ratio W/L would be 0.
0 = 0.3d + 11
Solving for d, we find:
0.3d = -11
d = -11/0.3
d ≈ -36.67
The value of d should be positive, so it is not possible for the bird to fly directly to its nesting area D.
(d) If the birds of a certain species reach the shore at a point 6 km from B, we can determine the ratio W/L by using the equation for the ratio W/L from part (b).
W/L = 0.3x + 11
Substituting x = 6 into the equation:
W/L = 0.3(6) + 11
W/L = 1.8 + 11
W/L = 12.8
Therefore, it takes a bird approximately 12.8 times more energy to fly over water than land.