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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
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.
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Single Variable Calculus: Early Transcendentals
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