For a fish swimming at a speed v relative to the water, the energy expenditure p

Question

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 the formula below, where a is the proportionality constant. Determine the value of v that minimizes E. (Note: This result has been verified experimentally.) 1 0

Explanation / Answer

dE/dv = [(v-u)*aL*3v^2-aLv^3]/(v-u)^2 To find the min, we need to set the first derivative=0 and solve for v. Since the denominator can never cause a fraction to precisely equal zero, we multiply both sides by the denominator (in other words, just remove the denominator from the equation). [(v-u)*aL*3v^2-aLv^3]/(v-u)^2=0 (v-u)*aL*3v^2-aLv^3=0 divide both sides by aL: 3(v-u)v^2 - v^3=0 Clearly if v=0, the equation is solved (but at v=0, nothing interesting happens, so we ignore that solution). So we assume v0, now let's divide by v^2: 3(v-u) - v=0 3v-3u-v=0 2v-3u=0 v=1.5u

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