Question

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 5 km from the nearest point B on a straight 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 13 km apart. (a) Suppose flying over water takes 1.4 times as much energy per kilometer as flying over land. Then how far should point C be from B to minimize the total energy the bird expends as it returns to its nesting area? 5.1 km (b) If the bird flies to the shore at point x kilometers from B on its trip from the island to the nest, determine the ratio of energy spent over water to energy spent over land. (Your answer should depend only on x.) W/L =

          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 5 km from the nearest point B on a straight 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 13 km apart.

(a) Suppose flying over water takes 1.4 times as much energy per kilometer as flying over land. Then how far should point C be from B to minimize the total energy the bird expends as it returns to its nesting area?
5.1 km

(b) If the bird flies to the shore at point x kilometers from B on its trip from the island to the nest, determine the ratio of energy spent over water to energy spent over land. (Your answer should depend only on x.)
W/L =

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Added by Dolores P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Jacob Fry

08/24/2022

Step 1

The distance it flies over water is then the hypotenuse of a right triangle with sides of 5 km (the distance from the island to point B) and x km (the distance from B to C). Using the Pythagorean theorem, this distance is sqrt(25 + x^2) km. Show more…

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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 5 km from the nearest point B on a straight 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 13 km apart. (a) Suppose flying over water takes 1.4 times as much energy per kilometer as flying over land. Then how far should point C be from B to minimize the total energy the bird expends as it returns to its nesting area? 5.1 km (b) If the bird flies to the shore at point x kilometers from B on its trip from the island to the nest, determine the ratio of energy spent over water to energy spent over land. (Your answer should depend only on x.) W/L =

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Transcript

00:01 In this question, we're told that we have a bird released from a point, which is, so here's the land.

00:10 This is land.

00:11 We've got the point b on land, which is five kilometres from where the bird is released.

00:21 The bird is released here.

00:26 And it flies to a point c on the shoreline and then flies along the shoreline to its nesting area.

00:35 D.

00:37 We're told that point b and d are 13 kilometers apart.

00:44 So part a, we want to find out how far is point b from point c to minimize the total energy? so if point b is a distance x from point c, then, and we're told also that the energy expended here, so e water equals 1 .4e land.

01:11 So the energy released going from a to c, eac is going to be a square root of 25 plus x squared times 1 .4e land, and the energy going from c to d is going to be 13 minus x times e land.

01:37 So the total energy divided by the energy going over the land, whatever that might be, is going to be 1 .4 times the square root of 25 plus x squared, plus 13 minus x.

01:54 And we want to minimize this with respect to x.

02:00 So we're going to do, let's call this combination e prime, or e bar, d, e bar by d x.

02:12 Is going to be zero when we're at the minimum energy configuration.

02:17 And this is going to be 1 .4x over root 25 plus x squared minus 1.

02:28 So this tells us that 25 plus x squared with the square root is 1 .4x.

02:35 Now squaring both sides, we get 25 plus x squared is 1 .4 squared x squared...

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