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 4 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 9 km apart. (Round your answers to two decimal places.) island (a) In general, if it takes 1.2 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? 10.55 X km from B (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. W L 1.2√ 49+x² 13-x x (c) What should the value of W/L be in order for the bird to fly directly to its nesting area D? x 0 (d) If the ornithologists observe that birds of a certain species reach the shore at a point 2 km from B, how many times more energy does it take a bird to fly over water than land? 1.58
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 4 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 9 km apart. (Round your answers to two decimal places.) island (a) In general, if it takes 1.2 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? 10.55 X km from B (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. W L 1.2√ 49+x² 13-x x (c) What should the value of W/L be in order for the bird to fly directly to its nesting area D? x 0 (d) If the ornithologists observe that birds of a certain species reach the shore at a point 2 km from B, how many times more energy does it take a bird to fly over water than land? 1.58
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Energy Expenditure and Flight Patterns of Birds**
Ornithologists have observed that some species of birds avoid flying over large bodies of water during daylight hours, generally favoring travel over land. This is primarily because more energy is required for the bird to fly over water than over land due to the difference in air currents and other environmental factors. A bird with these tendencies is released from an island 4 km from the nearest point B on the shoreline, then flies to a point C on the shoreline and subsequently along the shoreline to its nesting area D. The distance between points B and D is 9 km. It is assumed that the bird instinctively chooses a path that minimizes its energy expenditure. (All answers should be rounded to two decimal places.)
**Diagram Explanation:**
A diagram showing an island has been provided where:
- The island is 4 km away from point B on the shore.
- Point C is a chosen point on the shoreline.
- The distance from point B to point D along the shoreline is 9 km.
- The bird flies from the island to point C on the shore and then along the shore to point D.
**Questions and Solutions:**
* **(a) Optimal Path to Minimize Energy:**
- In general, if it takes \( 1.2 \) times as much energy to fly over water as over land, the bird should fly to a point \( C \) on the shoreline to minimize the total energy expended.
- Solving the problem results in minimizing the total energy expended, specifically:
\[
10.55 \text{ km from point } B.
\]
* **(b) Energy Expenditure Function:**
- 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, a function for the ratio \( \frac{W}{L} \) in terms of \( x \), the distance from \( B \) to \( C \), is given by:
\[
\frac{W}{L} = \frac{1.2\sqrt{49 + x^2}}{13 - x}.
\]
* **(c) Direct Flight Energy Expenditure:**
- To find the value of \( \frac{W}{L} \) for the bird to fly directly to its nesting area](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbeeb7c66-0798-43fa-9a0f-eb1dc3206797%2Fff7f2ca8-ed11-45a4-a04b-a0c488fca2f2%2Ffwepy2b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Energy Expenditure and Flight Patterns of Birds**
Ornithologists have observed that some species of birds avoid flying over large bodies of water during daylight hours, generally favoring travel over land. This is primarily because more energy is required for the bird to fly over water than over land due to the difference in air currents and other environmental factors. A bird with these tendencies is released from an island 4 km from the nearest point B on the shoreline, then flies to a point C on the shoreline and subsequently along the shoreline to its nesting area D. The distance between points B and D is 9 km. It is assumed that the bird instinctively chooses a path that minimizes its energy expenditure. (All answers should be rounded to two decimal places.)
**Diagram Explanation:**
A diagram showing an island has been provided where:
- The island is 4 km away from point B on the shore.
- Point C is a chosen point on the shoreline.
- The distance from point B to point D along the shoreline is 9 km.
- The bird flies from the island to point C on the shore and then along the shore to point D.
**Questions and Solutions:**
* **(a) Optimal Path to Minimize Energy:**
- In general, if it takes \( 1.2 \) times as much energy to fly over water as over land, the bird should fly to a point \( C \) on the shoreline to minimize the total energy expended.
- Solving the problem results in minimizing the total energy expended, specifically:
\[
10.55 \text{ km from point } B.
\]
* **(b) Energy Expenditure Function:**
- 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, a function for the ratio \( \frac{W}{L} \) in terms of \( x \), the distance from \( B \) to \( C \), is given by:
\[
\frac{W}{L} = \frac{1.2\sqrt{49 + x^2}}{13 - x}.
\]
* **(c) Direct Flight Energy Expenditure:**
- To find the value of \( \frac{W}{L} \) for the bird to fly directly to its nesting area
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