Question 1174975
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**