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Question 285794: The golden ratio. The ancient Greeks thought that the
most pleasing shape for a rectangle was one for which the
ratio of the length to the width was approximately 8 to 5,
the golden ratio. If the length of a rectangular painting is
2 ft longer than its width, then for what dimensions would
the length and width have the golden ratio?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Golden ratio is 8 to 5.
This is the ratio of the length to the width.
Length of the painting is 2 feet longer than its width.
This means that L = W+2
If the ratio of the length to the width if 8:5, this means that:
L/W = 8/5
Cross multiply to get:
5L = 8W
If L = W+2, you can substitute in this equation to get:
5*(W+2) = 8W
Simplify to get:
5W + 10 = 8W
Subtract 5W from both sides of this equation to get:
10 = 3W
Divide both sides of this equation by 3 to get:
W = 10/3 feet.
When W = 10/3 feet, L = W + 2 = 16/3
With L = 16/3 and W = 10/3, the ratio of L to W is equal to:
(16/3) / (10/3).
Multiply this faction by 3/3 and you get 16/10.
Multiply this fraction by (1/2)/(1/2) and you get:
8/5 which is the golden ratio.
The length and width will have the golden ration when the length = (16/3) and the width = (10/3).
Length of 16/3 is the same as 5.33333333.
Width of 10/3 is the same as 3.33333333.
Length is 2 feet longer than the width because 5.33333333 - 3.33333333 = 2.
ratio of Length to Width if 5.33333333/3.33333333 = 1.6 which is the same as 8/5.
That's your answer:
L = 5.33333333 = (16/3).
W = 3.33333333 = (10/3).
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