SOLUTION: One principle used by the ancient Greeks to get shapes that are pleasing to the eye in art and architecture was the Golden Rectangle. If a square is removed from one end of a Golde

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Question 316673: One principle used by the ancient Greeks to get shapes that are pleasing to the eye in art and architecture was the Golden Rectangle. If a square is removed from one end of a Golden Rectangle, the sides of the remaining rectangle are proportional to the original rectangle. So the length and width of the original rectangle satisfy L/W=W/L-W. An artist wants her paining to be in the shape of a golden rectangle. If the length of the painting is 36 inches, then what should be the width?
Answer by solver91311(24713) About Me  (Show Source):
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The ratio of the width to the length of a golden rectangle is . In other words:



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



The exact answer to the question can be found relatively easily by noting that the golden ratio, , is unique among positive numbers in that and since , we can say that (verification left to the student)

Hence, if then



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 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