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.

To minimize a function, means that we want to get the minimum value of the function.

The value of v, that minimizes E is:

The function is given as:

First, we differentiate E(v) with respect to E using the quotient rule.

Quotient rule states that:

Using the above rule, we have:

To minimize the function, we set the equation to 0

Cross multiply

Rewrite as:

Divide through by av^2L

Open brackets

Collect like terms

Divide both sides by 2

Hence, the value of v, that minimizes E is:

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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