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Roots and Coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

Target Six

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

Impedance Can Be Complex!

Age 16 to 18
Challenge Level

Part 1:

$P = IV$

I = $\frac{E}{Z_1 + Z_2} = \frac{E}{Z_1cos\theta + Z_2cos\phi + i(Z_1sin\theta + Z_2sin\phi)}$

|I| = $\frac{E}{\sqrt{(Z_1cos\theta + Z_2cos\phi)^2 + (Z_1sin\theta + Z_2sin\phi)^2}}$

The real power transferred to $Z_2$: P = I^2 R

P = $ I^2 Z_2cos \phi = \frac{E^2 Z_2cos \phi}{Z_1^2 + Z_2^2 + 2Z_1Z_2(cos \theta cos \phi + sin \theta \phi)}$= $ \frac{E^2 Z_2cos \phi}{Z_1^2 + Z_2^2 + 2Z_1Z_2cos( \theta - \phi )} $

Part 2:

To find the maximum power; we can differentiate the power expression with respect to $Z_2$ and set the derivative equal to zero.

$\frac{dP}{dZ_2} = \frac{d}{dZ_2}\frac{E^2 Z_2cos \phi}{Z_1^2 + Z_2^2 + 2Z_1Z_2cos( \theta - \phi )}$

Let U = $E^2 Z_2cos \phi$

Let V = $ Z_1^2 + Z_2^2 + 2Z_1Z_2cos( \theta - \phi ) $

$\frac{dP}{dZ_2} = \frac{VU' - UV'}{V^2}$

V' = $ 2Z_2 + 2Z_1 cos ( \theta - \phi)$

U' = $E^2 cos \phi$

Substituting thses values into the expression and setting it equal to zero we find that:

$Z_1 = Z_2$

For maximum power transfer between load and source we must therefore match the internal impedance of the load with the impedance of the source