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

Answer 1

The correct option is option C.

The width of the rectangular playground is
`-8+8\sqrt5` ft.

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= (16)/(1+\sqrt5)`

`\Rightarrow x= (16(1-\sqrt 5))/((1+\sqrt5)(1-\sqrt 5))` [ rationalize]

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

`\Rightarrow x= (16(1-\sqrt 5))/(1-5)`

`\Rightarrow x= (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