Question

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

Area=length times width


area=1=x times (x-1)

1=x(x-1)
1=x^2-x
0=x^2-x-1
using quadratic formula
for
0=ax^2+bx+c
$x= \frac{-b+/- \sqrt{b^2-4ac} }{2a}$
for
0=1x^2-1x-1
a=1
b=-1
c=-1

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

$x= \frac{1+ \sqrt{5} }{2}$ or $x= \frac{1- \sqrt{5} }{2}$
the 2nd one will be negative so we reject that because we can't have negative lengths

so
$x= \frac{1+ \sqrt{5} }{2}$
the length is $\frac{1+ \sqrt{5} }{2}$

Answer 2

Answer:

B because

Step-by-step explanation:

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?

and it is

1+ root 5 divided by 2