SOLUTION: 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 8 to 5, the golden ratio. If the length of a rec

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Question 125348: 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 8 to 5, the golden ratio. If the length of a rectangular painting is 2 feet longer than its width, then for what dimension would the length and width have the golden ratio?
Answer by solver91311(24713) About Me  (Show Source):
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I'm going to presume that you are expected to calculate this using 8 to 5 as an approximation of the golden ratio, even though the actual golden ratio is an irrational number expressed exactly by %281%2Bsqrt%285%29%29%2F2

Let x be the unknown width, and then x + 2 would be the unknown length. Since the sides must be in the proportion 8:5, we can write:

%28x%2B2%29%2Fx=8%2F5

Cross-multiplying:

5x%2B10=8x

5x-8x=-10

-3x=-10

x=10%2F3 giving us the width. The length is two feet more, so the length is 10%2F3%2B2=10%2F3%2B6%2F3=16%2F3

If you are curious, write back and I'll send you the calculations using the correct value for the golden ratio.