Concept explainers
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
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.]
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,
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
From Figure (1), it is noticed that there is one local minimum between
Point
The value 4606.875 is minimum for
Therefore, the value of v which minimizes the total energy is 7.5mi/h.
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