## USE THE GOLDEN RATIO!!!!!!!!!!! Suppose you want to use synthetic turf as the surface for a rectangular playground. The design ca

Question

USE THE GOLDEN RATIO!!!!!!!!!!!

Suppose you want to use synthetic turf as the surface for a rectangular playground. The design calls for a golden rectangle where the ratio of the longer length to the width is (1+√5) :2. If the longer length is 16 feet, which expression, in simplified form, represents the width of the playground?”

A. 8+8√5 ft

B. 16+16√5 /3 ft

C. −8+8√5 ft

D. 4√5+20 /5 ft

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2022-01-21T08:52:53+00:00
2022-01-21T08:52:53+00:00 1 Answer
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## Answers ( )

Answer:The correct option is option C.The width of the rectangular playground is[tex]-8+8\sqrt5[/tex]ft.Step-by-step explanation:Area of rectangular plot is = length × wide.

Given that,The ratio of longer length to the width of the rectangular playground is

(1+√5): 2

Let the length and width of the rectangular playground be (1+√5)x and 2x.

But the length of the longer side of the rectangular playground is = 16 feet.

According to the problem,(1+√5)x= 16

[tex]\Rightarrow x= \frac{16}{1+\sqrt5}[/tex]

[tex]\Rightarrow x= \frac{16(1-\sqrt 5)}{(1+\sqrt5)(1-\sqrt 5)}[/tex] [ rationalize]

[tex]\Rightarrow x= \frac{16(1-\sqrt 5)}{(1)^2-(\sqrt5)^2}[/tex] [ (a+b)(a-b)=a²-b²]

[tex]\Rightarrow x= \frac{16(1-\sqrt 5)}{1-5}[/tex]

[tex]\Rightarrow x= \frac{16(1-\sqrt 5)}{-4}[/tex]

[tex]\Rightarrow x=-4(1-\sqrt 5)}[/tex]

[tex]\Rightarrow x=-4+4\sqrt 5[/tex]

Then the width of the playground is = 2x[tex]=2(-4+4\sqrt5)[/tex] ft

[tex]=-8+8\sqrt5[/tex] ft