Question 1174975: Steve Gomez wishes to build a rectangular gaming arena for his buddies, Reyna and Justine. The area of the said gaming arena is 724 m^2. Its dimensions, length and width, are 1: φ, respectively. The length is equal to (2r − 1) m. Find the value of r, the width, and the perimeter of the gaming arena.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**1. Understand the Golden Ratio (φ)**
* The golden ratio (φ) is approximately 1.618. It's defined as (1 + √5) / 2.
* The dimensions are in the ratio 1:φ, meaning if the width is 'w', the length is 'wφ'.
**2. Use the Area Information**
* Area = Length * Width = 724 m²
* Let width = w. Then length = wφ.
* w * wφ = 724
* w²φ = 724
* w² = 724 / φ
* w = √(724 / φ)
**3. Calculate the Width (w)**
* w = √(724 / 1.618)
* w = √(447.466)
* w ≈ 21.153 m
**4. Calculate the Length (l)**
* Length (l) = wφ = 21.153 * 1.618
* l ≈ 34.235 m
**5. Use the Length Information to Find r**
* Length (l) = 2r - 1
* 34.235 = 2r - 1
* 35.235 = 2r
* r = 35.235 / 2
* r ≈ 17.6175
**6. Calculate the Perimeter (P)**
* Perimeter (P) = 2(Length + Width)
* P = 2(34.235 + 21.153)
* P = 2(55.388)
* P ≈ 110.776 m
**Results**
* **r ≈ 17.6175**
* **Width (w) ≈ 21.153 m**
* **Perimeter (P) ≈ 110.776 m**
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