A patio is shaped like a golden rectangle. Its length (the longer side) is 16 ft. What is the patio's width?
*Write your answer in simplified radical form.*
Answer (3)
0\\\\"> a g o l d e n rec t an g l e : a a + b = b a − − − − − − − − − − − − − − − − a + b = 16 [ f t ] ⇒ b = 16 − a a 16 = 16 − a a ⇔ 16 ( 16 − a ) = a 2 a 2 + 16 a − 256 = 0 ⇒ Δ = 1 6 2 − 4 ⋅ ( − 256 ) = 256 ⋅ 5 ⇒ Δ = 16 5 a 1 = 2 − 16 − 16 5 = − 8 − 8 5 < 0 a 2 = 2 − 16 + 16 5 = − 8 + 8 5 = 8 ( 5 − 1 ) > 0
a = 8 ( 5 − 1 ) ⇒ b = 16 − 8 ( 5 − 1 ) = 16 − 8 5 + 8 = 8 ( 3 − 5 ) A n s . T hi s g o l d e n r a c t an g l e ha s s i d es : . 8 ( 5 − 1 ) [ f t ] an d 8 ( 3 − 5 ) [ f t ]
The width of the patio is 16 × 2 5 − 16 × 2 1 feet, which simplifies to 8 5 − 8 feet.
To find the width of the patio, we need to use the properties of a golden rectangle. A golden rectangle is defined as a rectangle where the ratio of the length to the width is equal to the golden ratio, ϕ ,[/tex] which is approximately 1.61803398875. The golden ratio can also be expressed in terms of the square root of 5 as [tex] 2 1 + 5 .
Given that the length of the patio is 16 feet, we can set up the following proportion to find the width (w):
Width Length = 2 1 + 5
w 16 = 2 1 + 5
To solve for w, we cross-multiply:
2 × 16 = w × ( 1 + 5 )
32 = w + w 5
Now, we isolate the term with 5 :
w 5 = 32 − w
Next, we solve for w by dividing both sides by 5 :
w = 5 32 − 5 w
To simplify the expression, we rationalize the denominator by multiplying the numerator and denominator by 5 :
w = 5 32 5 − 5 w 5
Now, we have two expressions for w on the right side of the equation. To solve for w, we can express w in terms of itself:
w = 8 5 − 5 w
Multiplying through by 5 to eliminate the fraction:
5 w = 40 5 − w
Adding w to both sides:
6 w = 40 5
Dividing by 6:
w = 6 40 5
Simplifying the fraction:
w = 3 20 5
We can further simplify this expression by recognizing that the length of the patio is 16 feet, which is 3 48 feet. We want to express the width in terms of the length:
w = 3 48 × 2 5 − 3 48 × 2 1
w = 16 × 2 5 − 16 × 2 1
w = 8 5 − 8
Therefore, the width of the patio in simplified radical form is 8\sqrt{5} - 8 \) fee
The width of the patio, which is a golden rectangle with a length of 16 ft, is found to be 8 ( 5 − 1 ) feet in simplified radical form. This width maintains the golden ratio with the given length. This calculation incorporates the properties of the golden rectangle by setting up a proportional relationship based on the golden ratio.
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