To determine The value of v which minimizes the total energy required to swim a fixed distance. Answer The value of v which minimizes the total energy is 7.5mi/h. Explanation Given: The total energy required to swim a distance of 10 mi against a current of 5 mi/h is given by, E ( v ) = 2.73 v 3 10 v − 5 Calculation: Use a graphing calculator to draw the graph of the function E with respect to v as shown in Figure 1. Figure (1) Locate the point at which local minimum occurs to find the value of v which minimizes the total energy. The value E ( b ) is local minimum value if E ( b ) ≤ E ( v ) when b is near v . From Figure (1), it is noticed that there is one local minimum between v = 0 and v = 10 . Point ( 7.5 , 4606.875 ) is the lowest point. This point is the point of local minimum. The value 4606.875 is minimum for v = 7.5 . Therefore, the value of v which minimizes the total energy is 7.5mi/h.

Precalculus: Mathematics for Calculus - 6th Edition

6th Edition

ISBN: 9780840068071

Author: Stewart, James, Redlin, Lothar, Watson, Saleem

Publisher: Cengage Learning

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Textbook Question

Chapter 2.3, Problem 52E

Migrating Fish A fish swims at a speed v relative to the water, against a current of 5 mi/h. Using a mathematical model of energy expenditure, it can be shown that the total energy E required to swim a distance of 10 mi is given by

E ( v ) = 2.73 v 3 10 v − 5

Biologists believe that migrating fish try to minimize the total energy required to swim a fixed distance. Find the value of v that minimizes energy required.

[Note: This result has been verified; migrating fish swim against a current at a speed 50% greater than the speed of the current.]

Expert Solution & Answer

To determine

The value of v which minimizes the total energy required to swim a fixed distance.

Answer to Problem 52E

The value of v which minimizes the total energy is 7.5mi/h.

Explanation of Solution

Given:

The total energy required to swim a distance of 10 mi against a current of 5 mi/h is given by,

E(v)=2.73v310v−5

Calculation:

Use a graphing calculator to draw the graph of the function E with respect to v as shown in Figure 1.

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 2.3, Problem 52E

Figure (1)

Locate the point at which local minimum occurs to find the value of v which minimizes the total energy.

The value E(b) is local minimum value if E(b)≤E(v) when b is near v.

From Figure (1), it is noticed that there is one local minimum between v=0 and v=10 .

Point (7.5,4606.875) is the lowest point. This point is the point of local minimum.

The value 4606.875 is minimum for v=7.5 .

Therefore, the value of v which minimizes the total energy is 7.5mi/h.

Chapter 2.3, Problem 51E

Chapter 2.3, Problem 53E

Chapter 2 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

Ch. 2.1 - Prob. 1E Ch. 2.1 - For a function f, the set of all possible inputs...Ch. 2.1 - (a) Which of the following functions have 5 in...Ch. 2.1 - A function is given algebraically by the formula...Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - Prob. 9E Ch. 2.1 - Prob. 10E

Ch. 2.1 - Prob. 11E Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Prob. 14E Ch. 2.1 - Prob. 15E Ch. 2.1 - Prob. 16E Ch. 2.1 - Prob. 17E Ch. 2.1 - Prob. 18E Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - Prob. 21E Ch. 2.1 - Prob. 22E Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - Prob. 25E Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Prob. 30E Ch. 2.1 - Prob. 31E Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Prob. 34E Ch. 2.1 - Prob. 35E Ch. 2.1 - Prob. 36E Ch. 2.1 - Prob. 37E Ch. 2.1 - Prob. 38E Ch. 2.1 - Prob. 39E Ch. 2.1 - Prob. 40E Ch. 2.1 - Prob. 41E Ch. 2.1 - Prob. 42E Ch. 2.1 - Prob. 43E Ch. 2.1 - Prob. 44E Ch. 2.1 - Prob. 45E Ch. 2.1 - Prob. 46E Ch. 2.1 - Prob. 47E Ch. 2.1 - Prob. 48E Ch. 2.1 - Prob. 49E Ch. 2.1 - Prob. 50E Ch. 2.1 - Prob. 51E Ch. 2.1 - Prob. 52E Ch. 2.1 - Prob. 53E Ch. 2.1 - Prob. 54E Ch. 2.1 - Prob. 55E Ch. 2.1 - Prob. 56E Ch. 2.1 - Prob. 57E Ch. 2.1 - Prob. 58E Ch. 2.1 - Prob. 59E Ch. 2.1 - Prob. 60E Ch. 2.1 - Prob. 61E Ch. 2.1 - Prob. 62E Ch. 2.1 - Prob. 63E Ch. 2.1 - Prob. 64E Ch. 2.1 - Prob. 65E Ch. 2.1 - Prob. 66E Ch. 2.1 - Prob. 67E Ch. 2.1 - Prob. 68E Ch. 2.1 - Prob. 69E Ch. 2.1 - Prob. 70E Ch. 2.1 - Prob. 71E Ch. 2.1 - Prob. 72E Ch. 2.1 - Prob. 73E Ch. 2.1 - Prob. 74E Ch. 2.1 - Relativity According to the Theory of Relativity,...Ch. 2.1 - Income Tax In a certain country, income tax T is...Ch. 2.1 - Internet Purchases An Internet bookstore charges...Ch. 2.1 - Cost of a Hotel Stay A hotel chain charges 75 each...Ch. 2.1 - Speeding Tickets In a certain state the maximum...Ch. 2.1 - Height of Grass A home owner mows the lawn every...Ch. 2.1 - Prob. 81E Ch. 2.1 - Prob. 82E Ch. 2.1 - Prob. 83E Ch. 2.1 - Prob. 84E Ch. 2.1 - Prob. 85E Ch. 2.2 - Prob. 1E Ch. 2.2 - Prob. 2E Ch. 2.2 - Prob. 3E Ch. 2.2 - Match the function with its graph. 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Solution Summary: The author explains that the value of v which minimizes the total energy required to swim a fixed distance is 7.5mi/h.