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:

length of golden rectangle = 1.618

Step-by-step explanation:

It is given that the length (L) of golden rectangle is x

And width (W) of rectangle is x-1

so we have

L = x

W= x-1

Now we have area of rectangle given by

Area = L × W

Area = x(x-1)                             ( we plug L= x and W= x-1)

It is given that area of rectangle is 1 square unit.

So we have

$x(x-1)=1$

$x^2 -x=1$       ( distribute x and remove parenthesis )

$x^2 -x-1=0$    ( subtract 1 from both sides )

Now to solve for x we need to use the quadratic formula

For quadratic equation $ax^2 +bx+c=0$

The quadratic formula is given by $x=\frac{-b\pm\sqrt{b^2-4ac} }{2a}$

Here we have a= 1 , b=-1 c=-1 , so we have

$x=\frac{-(-1)\pm\sqrt{(-1)^2 -4(1)(-1) }}{2(1)} \\x=\frac{1\pm\sqrt{1+4} }{2} \\x=\frac{1\pm\sqrt{5} }{2}$

$x= \frac{1\pm2.236}{2} \\x=\frac{1+2.236}{2}$  or  $x=\frac{1-2.236}{2}$

$x= 1.618$ or x=-0.618

We can not have length negative

hence

length of golden rectangle = 1.618

Answer 2

The answer is b. I just did the question and that is the answer