Related Rates

[tangents]

Hey Guys, I learning about Related rates and although I understand the basic concepts, I'm stuggling with this problem

- When two resistors r1 and R2 are connected in parallel, the total resistance R is given by the equation 1/R=1/R1+1/R2. If R1 and R2 are increasing at rates of .01 ohm/sec and .02 ohm/sec respectively, at what rate is R changing at the instant R1=30 ohms and R2= 90 ohms?

My Work:

dr1/dt= .01
dr2/dt= .02
dr/dt= Trying to find this when r1=30 and r2=90
First what I did was plug in R1 and R2 to get total resistance which i got to be 1/22.5. Next I took the derivative of that equation: -1/r1²(dr/dt)=-1/r²(dr1/dt)-1/r²2(dr/dt). I plugged in R1 and R2 along with dr1/dt and dr2/dt along with the total resistance and tried to solve for dr/dt (total resistance change). But I end up with .018 ohms/sec whereas the answer the book shows is 11/1600 or .006 ohm/sec.

Where did i go wrong? Could someone please show me a easy to follow step by step in getting the answer ^_^, i would greatly appreciate it!.

 

[royhaas]

I think you just have an arithmetic problem somewhere. I get
\(\displaystyle dR/dt = (8100/16)(.01/900+.02/8100)=.11/16=11/1600\).

 

[DistantBeliefs]

where/how did you get 8100/16? i found Rtotal to be 22.5 so don't you just square that and multiply it with the fraction?

 

[Gene]

The derivitive is
-1/r²*dr/dt=-1/r₁²dr₁dt-1/r₂²dr₂/dt
That gives
dr/dt=(r/r₁)²dr₁dt+(r/r₂)²dr₂/dt
dr/dt = (22.5/30)²*.01+(22.5/90)²*.02
That gives the right answer.

I'm not sure what you were doing with
-1/r1²(dr/dt)=-1/r²(dr1/dt)-1/r²2(dr/dt)
Something should have happened to the signs when you were moving them accross the =. Like
1/r1²(dr/dt)=1/r²(dr1/dt)-1/r²2(dr/dt)
and I don't follow the denominator movements.
Since the flip-flop was neither needed nor desired I'll leave it at that.