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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

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 over 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.

  1. (a) In general, if it takes 1.4 times as much energy to fly over water as it does over land, to what point C should the bird fly in order to minimize the total energy expended in returning to its nesting area?
  2. (b) Let W and L denote the energy(in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio W/L mean in terms of the bird’s flight? What would a small value mean? Determine the ratio W/L corresponding to the minimum expenditure of energy.
  3. (c) What should the value of W/L be in order for the bird to fly directly to its nesting area D? What should the value of W/L be for the bird to fly to B and then along the shore to D?
  4. (d) If the ornithologists observe that birds of a certain species reach the shore at a point 4 km from B, how many times more energy does it take a bird to fly over water than over land?

Chapter 4.7, Problem 81E, Ornithologists have determined that some species of birds tend to avoid flights over large bodies of

(a)

To determine

To find: The position of C in order for minimizing the total energy expended.

Explanation

Given:

|BD|=13

Distance between the island and the nearby shoreline=5km.

Formula used:

Pythagoras Theorem: If c denotes the length of the hypotenuse and a and b denote the length of the other two sides of a right angle triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation: a2+b2=c2.

Calculation:

Let the energy consumed for flying per km over land be k.

The energy consumed for flying per km over water is 1.4 times of the energy consumed for flying per km over land.

Hence, the energy consumed for flying per km over water is 1.4k.

In Figure 1, C is located x km far from the point B.

Then, C is (13x) km far from the point D.

Hence, the bird flies (13x) km over land.

The energy consumed for flying per km over land is k(13x).

By Pythagorean Theorem on the triangle ΔABC,

AC=52+x2=x2+25

Hence, the bird flies x2+25 km over water.

The energy consumed for flying per km over water is 1.4kx2+25.

So, the total energy is,

E=1.4kx2+25+k(13x).

Differentiate E with respect to x,

E=1.4k×12×2xx2+25k=1.4kxx2+25k

For critical points,

E=01.4kxx2+25k=01

(b)

To determine

To find: The ratio W/L corresponding to the minimum expenditure of energy.

(c)

To determine

To find: The ratio W/L corresponding to the minimum expenditure of energy.

(d)

To determine

To find: The ratio of the energy of the bird to fly over water than the energy of the bird to fly over land.

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