Question 316673
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The ratio of the width to the length of a golden rectangle is *[tex \Large \varphi\ =\ \frac{1\ +\ \sqrt{5}}{2}\ \approx\ 1.618].  In other words:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ l\ =\ w\varphi]


but since we are given the length, we need to solve


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ w\ =\ \frac{l}{\varphi}]


The exact answer to the question can be found relatively easily by noting that the golden ratio, *[tex \Large \varphi], is unique among positive numbers in that *[tex \Large \frac{1}{\varphi}\ =\ \varphi\ -\ 1] and since *[tex \Large \varphi\ =\ \frac{1\ +\ \sqrt{5}}{2}], we can say that *[tex \Large \ \varphi\ -\ 1\ =\ \frac{-1\ +\ \sqrt{5}}{2}] (verification left to the student)


Hence, if *[tex \Large l\ =\ 36] then


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ w\ =\ 18\left(-1\ +\ \sqrt{5}\right)\text{ in}]


Which is the simplest form of the exact answer.


However, from a practical point of view, what is wanted is a dimension, to the nearest inch of an approximation of the width for a golden rectangle with a length of 36 inches.


Just multiply *[tex \Large 36 \times 0.62] and round to the nearest inch.  Why to the nearest inch?  Because your only measurement was given to the nearest inch and you should never have the result of a calculation based on a measurement expressed to a greater precision than the least precise of your input measurements.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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