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 e(v) = av3 · l v − u where a is the proportionality constant. (a) determine the value of v that minimizes

e. note: this result has been verified experimentally.

Answer 1

For
$e(v)=\dfrac{alv^{3}}{v-u}$
the derivative with respect to v is
$e'(v)=al\dfrac{(v-u)\cdot3v^{2}-v^{3}}{(v-u)^{2}}$
We can set this to zero and remove several common factors to get
$0=2v-3u\\\\v=\frac{3}{2}u$

The value of v that minimizes e(v) is v = (3/2)u.