Optimizing Fish Swimming Energy: Finding Minimal Expenditure

[girlygirl93]

The energy required by a fish swimming at speed v to travel a distance L>0 in a current of speed u>0 is given by

E(v) = aL((v^3)/(v-u)), v>u

where a>0 is a proportionality constant.
a) Find the speed of the fish which results in minimal energy expenditure.
b) Give a qualitative sketch of the energy as a function of the speed of the fish.

I know that I am supposed to isolate v from the equation by using another equation in order to differentiate it, but I don't know how to. As well as a is a constant, and all values are positive. Also when I am drawing the sketch will the value a still be included? How do I include this in a graph? How do I make an equation that involves both L and u in order to differentiate in terms of v?

 

[SammyS]

The energy required by a fish swimming at speed v to travel a distance L>0 in a current of speed u>0 is given by

E(v) = aL((v^3)/(v-u)), v>u

where a>0 is a proportionality constant.
a) Find the speed of the fish which results in minimal energy expenditure.
b) Give a qualitative sketch of the energy as a function of the speed of the fish.

I know that I am supposed to isolate v from the equation by using another equation in order to differentiate it, but I don't know how to. As well as a is a constant, and all values are positive. Also when I am drawing the sketch will the value a still be included? How do I include this in a graph? How do I make an equation that involves both L and u in order to differentiate in terms of v?

Work this out for a given set of values of a, L, and u. In other words, treat a, L, and u as constant values. In general, how do you find min/max values for E(v) ?

For the graph: It wants a qualitative graph, not quantitative.

Define a 'relative' velocity variable, perhaps call it r. Let r = v/u . Then v = ur. Plug that in for v, and see what you get.​

 

[girlygirl93]

That makes sense but for the graph I don't understand how to plot it with so many variables involved. Wouldn't introducing another variable just make it more complicated?

 

[girlygirl93]

** sorry not variables, I meant different constants that are not numerical.

 

[Dick]

That makes sense but for the graph I don't understand how to plot it with so many variables involved. Wouldn't introducing another variable just make it more complicated?

If v is just a little bit bigger than u, then E is very large. If v is very large then E is very large. At the critical point E is a minimum. What is the critical point? Just sketch a curve indicating that. But first find the critical point in terms of u. Do that first.