   Chapter 4.7, Problem 46E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v3. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current u (u < v), then the time required to swim a distance L is L/(v − u) and the total energy E required to swim the distance is given by E ( v ) = a v 3 ⋅ L v − u where a is the proportionality constant.(a) Determine the value of v that minimizes E.(b) Sketch the graph of E.Note: This result has been verified experimentally; migrating fish swim against a current at a speed 50% greater than the current speed.

(a)

To determine

To find: the value of v that minimizes E.

Explanation

Explanation;

Given:

The speed of the fish is v in the water.

The energy per unit time is proportional to v3.

The speed of the current is u.

The time required to swim a distance L is Lvu.

The total energy required to swim the distance is given by, E(v)=aLv3vu

Calculation:

The total energy required to swim the distance is given by,

E(v)=aLv3vu

Differentiate E with respect to x,

dEdv=aL(vu)3v2v3(vu)2

For critical points, dEdv=0

aL(vu)3v2v3(vu)2=03v33v2uv3=02v3=3v2uv=32u

Differentiate dEdv with respect to x,

d2Edv2=

(b)

To determine

To sketch: The graph of E.

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