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

Answer 1

Answer:

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