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+1
401
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Hi guys, I'm not 100% sure what to do with the u (intensity of happiness) in this problem. Could someone give me a hint as to how I should go about this question?

mathmeme  May 9, 2017
 #1
avatar+92777 
+3

Hi mathmeme

 

\(\mu\quad and \quad C \quad \text{are both constants}\\ Let \quad U(\theta) \text{ be referred to as U}\)

 

So we have:

 

\(U=\frac{\mu C}{\mu sin(\theta)+cos(\theta) }\\ U=\mu C(\mu sin(\theta)+cos(\theta) )^{-1}\\ \frac{dU}{d\theta}=-\mu C(\mu sin(\theta)+cos(\theta) )^{-2}(\mu cos(\theta)-sin(\theta))\\ \frac{dU}{d\theta}=\frac{-\mu C(\mu cos(\theta)-sin(\theta))}{(\mu sin(\theta)+cos(\theta) )^{2}}\\ \text{When }U'=0\\ -\mu C(\mu cos(\theta)-sin(\theta))=0\\ \mu cos(\theta)=sin(\theta)\\ \mu=tan(\theta)\)

 

Trouble is this could give a minimum or a maximum.

 

I did a graph in Desmos and found that  mu = tan(theta)  gives both the minimum and the maximum degree of happiness.

Here is the graph:

 

https://www.desmos.com/calculator/1c8zst7hlo

 

Feel free to question me about this answer :)

Melody  May 9, 2017
 #3
avatar+62 
0

Thank you so much Melody! You're always so helpful :)

mathmeme  May 10, 2017
 #2
avatar+26750 
+2

Here's another graph to illustrate the point:

 

.Seems odd that the "maximum" has a lower value than the "minimum"!!

.

Alan  May 9, 2017
 #4
avatar+62 
0

Thank you for showing me the graph!

mathmeme  May 10, 2017

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