A golden rectangle has sides of length 1 and x. When a square with side length 1 is removed from the golden rectangle, the remaining rectangle has the same proportions as the original. Solve for x.
To solve for x, we can use the properties of a golden rectangle. A golden rectangle is a rectangle whose sides are in the ratio of approximately 1.618:1, which is also known as the golden ratio (denoted by phi, Φ).
Let's consider the given information: the original golden rectangle has sides of length 1 and x. When a square with a side length of 1 is removed, the remaining rectangle should have the same proportions.
The proportion of the original golden rectangle can be expressed as:
1 / x = x / (x - 1)
To solve this equation, we can cross-multiply:
x^2 - x = 1
Rearranging the equation:
x^2 - x - 1 = 0
This is a quadratic equation in terms of x. We can solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -1, and c = -1. Plugging these values into the quadratic formula:
x = (1 ± √((-1)^2 - 4(1)(-1))) / (2(1))
Simplifying further:
x = (1 ± √(1 + 4)) / 2
x = (1 ± √5) / 2
Therefore, the two possible values for x are approximately:
x ≈ (1 + √5) / 2 ≈ 1.61803 (the positive value)
x ≈ (1 - √5) / 2 ≈ -0.61803 (the negative value)
Since negative values do not make sense in the context of the problem, the value of x for a golden rectangle is approximately 1.61803.