Question

A "golden rectangle" is a rectangle where the ratio of the longer side to the shorter side is the "golden ratio." These rectangles are said to be visually pleasing. An example of a "golden rectangle" has a length equal to x units and a width equal to x – 1 units. Its area is 1 square unit. What is the length of this golden rectangle?

Answer 1

Answer:

Step-by-step explanation:

Approximately equal to a 1:1.61 ratio, the Golden Ratio can be illustrated using a Golden Rectangle. This is a rectangle where, if you cut off a square (side length equal to the shortest side of the rectangle), the rectangle that's left will have the same proportions as the original rectangle

Answer 2

The length of this golden rectangle is 1.618 units.

Given,

An example of a "golden rectangle" has,

length = x units,

width = (x - 1) units, and,

area = 1 unit².

To find,

Length of this golden rectangle.

Solution,

A golden rectangle is defined as a rectangle whose ratio of length to width is the "golden ratio".

Here it is given that, for the golden rectangle,

length = x units,

width = (x - 1) units,

area = 1 unit².

The area of this rectangle will be

$A=x(x-1)$

It is given to be 1 unit². So,

$A=x(x-1) = 1$

Simplifying the above equation, we get,

$x^2-x-1=0$

Solving the above quadratic equation using the quadratic formula.

Here,

$a=1, b=-1, c=-1$

$x=\frac{-(-1) \pm \sqrt{(-1)^2-4(1)(-1)} }{2}$

$\implies x=\frac{1 \pm \sqrt{1+4} }{2}$

$\implies x=\frac{1 \pm \sqrt{5} }{2}$

$\implies x=\frac{1 + \sqrt{5} }{2}=1.618, and \hspace{3} x=\frac{1 - \sqrt{5} }{2}=-0.618$

As length cannot be negative, so the only first value will be considered, that is,

$x=1.618$ units.

⇒ length of the rectangle = 1.618 units.

Therefore, the length of this golden rectangle is 1.618 units.

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