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

The value of v =
`(3u)/(2)` that minimize E.

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) = (av^(3)L)/(v - u)` , where a is a proportionality constant.

Now, for E(v) to be minimum the condition is
`(dE(v))/(dv) = 0`

⇒
`aL(d)/(dv)[(v^(3))/(v - u) ] = 0`

⇒
`aL[(3v^(2)(v - u) - v^(3) )/((v - u)^(2) ) ] = 0`

⇒ 3v³ - 3v²u - v³ = 0

⇒ 2v³ = 3v²u

⇒ v =
`(3u)/(2)`

Therefore, the value of v =
`(3u)/(2)` that minimizes E. (Answer)