Respuesta :
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
