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 feet longer than its width, then for what dimensions would the length and width have the golden ratio?
let x = the width
x+2 = the length
(x+2)/x = 8/5
To find the dimensions of the rectangular painting that would have the golden ratio, we need to set up an equation based on the given information.
Let's assume the width of the painting is "x" feet. According to the problem, the length of the painting is 2 feet longer than the width, so the length would be "x + 2" feet.
The golden ratio is approximately 8 to 5, which means the ratio of the length to the width should be 8/5. Therefore, we can set up the following equation:
(x + 2) / x = 8/5
To solve this equation, we can cross-multiply:
5(x + 2) = 8x
Now, distribute the 5:
5x + 10 = 8x
Next, subtract 5x from both sides:
10 = 3x
Finally, divide both sides by 3:
x = 10/3
So, the width of the rectangular painting is approximately 3.33 feet. And since the length is 2 feet longer, the length would be:
x + 2 = 10/3 + 2 = 10/3 + 6/3 = 16/3
Therefore, the dimensions that would have the golden ratio are approximately 3.33 feet by 5.33 feet.