Finding the variable by differentiating

Heys, does anyone know how to perform this question to check if I did mt working out correctly? The question is ...

Consider the flow of air past an ideal turbine with a swept area of A.

The fraction of power that can be extracted from the power contained in the incoming wind is called the ‘power coefficient,’ Cp = 4a(1‐a)^2. This is effectively the aerodynamic efficiency of the blades turbine in extracting power out of the incoming wind.

(b) Find the maximum value of Cp and find out at what value of ‘a’ this occurs. Hint differentiate with respect to ‘a’.

Working:

Finding the extremum (max or min) of a function means taking the derivative and setting it to zero

dCp/da = 4(1-a)^2 + 8a(1-a)(-1) =0

4(1-a)^2 = 8a (1-a)

4(1-a) =8a

1-a = 2a or a = 1/3

the max value occurs at a = 1/3?

Therefore, Cp = 16/27?

Thanks

Re: Finding the variable by differentiating

You are correct, but just one clarification. You divided by 1 - a, which gives another root to the equation dCp/da = 0, i.e., a = 1. You need to make sure that that extremum is not a maximum. Equally, you should show that the extremum at a = 1/3 *is* a maximum (e.g., because the derivative changes sign from + to -).

Re: Finding the variable by differentiating

How would I how that a = 1/3 is a maximum? I get what you mean though

Re: Finding the variable by differentiating

As I said, you can use the first derivative test, which requires that the first derivative changes sign from + to - at the critical point, or the second derivative test, which requires that the second derivative is negative.

Re: Finding the variable by differentiating

As **emakarov** points out, dividing the equation by (1-a) is a bad idea. In this case you lose one of your solutions.

Rather than dividing 4(1-a)^2 = 8a (1-a) by (1-a),

factor (1-a) from the left side of 4(1-a)^2 + 8a(1-a)(-1) = 0

You get (1-a){4(1-a)-8a} = 0, which can be simplified. This gives two solutions for a.

Take it from there.