# Sketching the graph of a swimming fish's energy

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Any help is greatly appreciated; I found this problem pretty frustrating. I replaced the "less than" symbol with the words "less than" because the computer seemed to have a hard time recognizing the symbol.

"For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v^3. 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 less than 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)=av^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.

Author's Note: This result has been verified experimentally; migrating fish swim against a current at a speed of 50% greater than the current speed."

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##### Solution Summary

Tips are provided as to how to perform thacalculations that allow for the graph of a swimming fish's energy to be plotted. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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E(v)=av^3(L/[v-u])

Differentiate E with respect to v and equate to zero to get the minimum value of E.

dE/dv = a*L*[ v^3*{-1/(v-u)^2} + 3v^2/(v-u) ] = ...

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