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)
Determine the value of v that minimizes E. (Note: This result has been verified experimentally.)

Answer 1

Answer:

The value of v = $\frac{3u}{2}$ that minimize E.

Step-by-step explanation:

The function that gives the energy lost by a fish E(v) moving with a velocity v against the water velocity u up to a distance L is given by

$E(v) = \frac{av^{3}L}{v - u}$ , where a is a proportionality constant.

Now, for E(v) to be minimum the condition is $\frac{dE(v)}{dv} = 0$

⇒ $aL\frac{d}{dv}[\frac{v^{3}}{v - u} ] = 0$

⇒ $aL[\frac{3v^{2}(v - u) - v^{3} }{(v - u)^{2} } ] = 0$

⇒ 3v³ - 3v²u - v³ = 0

⇒ 2v³ = 3v²u

⇒ v = $\frac{3u}{2}$

Therefore, the value of v = $\frac{3u}{2}$ that minimizes E. (Answer)