team15

Team 15 Casey Sweetser Glenn Radliff David Yeh 34. For a fish swimming at a speed "v" relative to the water, the energy expenditure per unit time is proportional to v^3. 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) and the total energy E required to swim the distance is given by

E(v)=av^3*(L)/(v-u)
where a is the proportionality constant.

a) Determine the value of v that minimizes E.
b) Sketch the graph of E.

Note: This result has been verified experimentally; migrating fish swim against a current at a speed 50% greater than the current speed.

E(v)=av^3*(L)/(v-u)
E(v)=aLv^3/(v-u)

           (v-u)*(3aLv^2)-(aLv^3)
E'(v)=___________________________
           (v-u)^2

       2*au^3*L-3av^2 *u^2
0=________________________
        v^2-2vu+u^2

v=3u/2

graph of E(v) since a is a constant, we set to one. L is a constant so we set to 1.