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

The correct option is option C.

The width of the rectangular playground is  $$-8+8\sqrt5$$ 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

$$\Rightarrow x= \frac{16}{1+\sqrt5}$$

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

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

$$\Rightarrow x= \frac{16(1-\sqrt 5)}{1-5}$$

$$\Rightarrow x= \frac{16(1-\sqrt 5)}{-4}$$

$$\Rightarrow x=-4(1-\sqrt 5)}$$

$$\Rightarrow x=-4+4\sqrt 5$$

Then the width of the playground is = 2x

$$=2(-4+4\sqrt5)$$ ft

$$=-8+8\sqrt5$$ ft