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avatar+665 

\(\frac{l+w}{l}=\frac{l}{w}\)

 

Find \(\frac{l}{w}\), your answer should be the golden ratio that is a fraction with radicals in simplest form.

 

(BTW, the answer is definitely NOT \(\frac{l+w}{l}\))

 

This is a fun math problem I solved yesterday night.

 Apr 28, 2019
edited by CalculatorUser  Apr 28, 2019
 #1
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multiply both by lw

l2+w2=l2​^w

subtract l2

w2=w

subtract w

w=0

??

 Apr 29, 2019
 #2
avatar+665 
+3

You incorrectly multiplied by lw,

 

it should be lw+w^2=l^2

 

You also CAN'T subtract l^2

CalculatorUser  Apr 29, 2019
 #3
avatar+22550 
+2

Easy Golden Ratio Quadratic

\(\dfrac{l+w}{l}=\dfrac{l}{w}\)

 

I assume:

\(\text{Let }\textbf{Golden Ratio} =\varphi\)

 

\(\begin{array}{|rcll|} \hline \dfrac{l+w}{l} &=& \dfrac{l}{w} \\\\ \dfrac{l}{l}+ \dfrac{w}{l} &=& \dfrac{l}{w} \\\\ 1 + \dfrac{w}{l} &=& \dfrac{l}{w} \quad | \quad \dfrac{l}{w} = \varphi,\ \dfrac{w}{l} = \dfrac{1}{\varphi} \\\\ 1 + \dfrac{1}{\varphi} &=& \varphi \quad |\quad \cdot \varphi \\\\ \varphi + 1 &=& \varphi^2 \\ \varphi^2 -\varphi - 1 &=& 0 \\\\ \varphi &=& \dfrac{1\pm \sqrt{1-4\cdot (-1) }}{2} \\ \varphi &=& \dfrac{1\pm \sqrt{5}}{2} \\\\ \mathbf{\varphi} &\mathbf{=}& \mathbf{\dfrac{1+ \sqrt{5}}{2}} \\ \hline \end{array}\)

 

 

laugh

 Apr 29, 2019

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