Related Rates setup issue

**tubasoldier** · February 23rd 2009, 11:38 AM

I'm having issues setting up a related rates problem. Mostly because it is talking about electricity and I'm not all that sure about how to visualize it.

The impedance Z(ohms) in a series circuit is related to the resistance R(ohms) and reactance X(ohms) by the equation $\sqrt[]{R^2+X^2}$ . If R is increasing at 3 ohms/sec and X is decreasing at 2 ohms/sec, at what rate is Z changing when R=10 ohms and X=20 ohms?

If I could just have some help with setting this up I would be very grateful.
Thanks

**Soroban** · February 23rd 2009, 01:32 PM

Hello, tubasoldier!

The subject doesn't matter ... whether it's electricity or fermentation of beer.
It's already set up . . . go for it!

Given: . $Z \:=\:\sqrt{R^2+X^2},\quad \frac{dR}{dt} \,=\,3,\quad \frac{dX}{dt} \,=\,-2$

Find $\frac{dZ}{dt}$ when $R = 10,\;x = 20$

We have: . $Z \;=\;\left(R^2+X^2\right)^{\frac{1}{2}}$

Differentiate with respect to time: . $\frac{dZ}{dt} \;=\;\frac{1}{2}\left(R^2+X^2\right)^{-\frac{1}{2}}\left(2R\!\cdot\!\frac{dR}{dt} + 2X\!\cdot\!\frac{dX}{dt}\right)$

. . and we have: . $\frac{dZ}{dt} \;=\;\frac{R\!\cdot\!\frac{dR}{dt} + X\!\cdot\!\frac{dX}{dt}}{\sqrt{R^2 + X^2}}$

Now plug in the given values . . .

**skeeter** · February 23rd 2009, 01:48 PM

Originally Posted by Soroban

The subject doesn't matter ... whether it's electricity or fermentation of beer.

I'll drink to that!

**tubasoldier** · February 23rd 2009, 03:50 PM

Originally Posted by Soroban

Hello, tubasoldier!

The subject doesn't matter ... whether it's electricity or fermentation of beer.

Thats pretty funny. You are currently king of my world. Thank you good sir!

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