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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Charlie 4 months 2022-01-21T08:52:53+00:00 1 Answer 0 views 0

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    2022-01-21T08:54:17+00:00

    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

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