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Re: Re: Solve given that a variable is between 0 and 1

• To: mathgroup at smc.vnet.net
• Subject: [mg52688] Re: Re: Solve given that a variable is between 0 and 1
• From: "Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk>
• Date: Thu, 9 Dec 2004 20:22:45 -0500 (EST)
• References: <cp3spf$98g$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

I didn't even try a manual solution because I assumed the fractional powers 
would frustrate the derivation. How wrong I was.

I now get the following analytic result:

((OverBar[y]*(w/r)^(b/(1 - b)))/(a*(1 + (w/r)^(b/(1 - b)))))^(1/b)

In fact, the derivation goes through so cleanly (i.e. all the powers match 
up at just the right points in the derivation) that it makes you wonder 
where the original equation came from!

Steve Luttrell

"yehuda ben-shimol" <benshimo at bgu.ac.il> wrote in message 
news:cp3spf$98g$1 at smc.vnet.net...
>
> The problem at hand is easily solved by hand in about 60 seconds.
> Yehuda
>
> -----Original Message-----
> From: Steve Luttrell [mailto:steve_usenet at _removemefirst_luttrell.org.uk]
To: mathgroup at smc.vnet.net
> Subject: [mg52688]  Re: Solve given that a variable is between 0 and 1
>
> To solve algebraically you need to give a numerical value to b, and then
> only certain values lead to simple closed-form solution. Otherwise, you 
> have
>
> to solve numerically.
>
> Algebraic example:
>
> b = 1/2;
> Solve[w - k^(1 - b)*r*((-((a*k^b - OverBar[y])/a))^
>       (1/b))^(-1 + b) == 0, k]
>
> gives
>
> {{k -> (w^2*OverBar[y]^2)/(a^2*(-r + w)^2)},
>  {k -> (w^2*OverBar[y]^2)/(a^2*(r + w)^2)}}
>
> Numerical example:
>
> b = 0.4;
> w = 1;
> a = 1;
> r = 1;
> OverBar[y] = 1;
> NSolve[w - k^(1 - b)*r*((-((a*k^b - OverBar[y])/a))^
>       (1/b))^(-1 + b) == 0, k]
>
> gives
>
> {{k -> 0.17677669529663688110021109052621}}
>
> Steve Luttrell
>
> "Jason" <jsr123 at gmail.com> wrote in message
> news:cos13d$dl2$1 at smc.vnet.net...
>> Hi all,
>>
>> I need some help with solving the following:
>>
>> \!\(Solve[
>>    w - k\^\(1 - b\)\ r\ \((\((\(-\(\(a\ k\^b -
>> y\&_\)\/a\)\))\)\^\(1\/b\))\)\
>> \^\(\(-1\) + b\) \[Equal] 0, k]\)
>>
>> given that a>0 and 0<b<1.
>>
>> How do I do this?
>>
>> Thanks MUCH!!---jason
>>
>
>
>