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. E (v) = av^3 L / (v - u) 1. Determine the value of v that minimizes E.

Question

Mathematics

Johnathan Joyce

17 September, 19:20

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. E (v) = av^3 L / (v - u) 1. Determine the value of v that minimizes E.

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Kathy Townsend · Answer 1

Value of v that minimizes E is v = 3u/2

Step-by-step explanation:

We are given that;

E (v) = av³L / (v-u)

Now, using the quotient rule, we have;

dE/dv = [ (v-u) •3av²L - av³L (1) ] / (v - u) ²

Expanding and equating to zero, we have;

[3av³L - 3av²uL - av³L] / (v - u) ² = 0

This gives;

(2av³L - 3av²uL) / (v-u) ² = 0

Multiply both sides by (v-u) ² to give;

(2av³L - 3av²uL) = 0

Thus, 2av³L = 3av²uL

Like terms cancel to give;

2v = 3u

Thus, v = 3u/2