document.write( "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. \n" ); document.write( "
Algebra.Com's Answer #850628 by CPhill(1959)\"\" \"About 
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Let's break down this problem step-by-step.\r
\n" ); document.write( "\n" ); document.write( "**1. Understand the Golden Ratio (φ)**\r
\n" ); document.write( "\n" ); document.write( "* The golden ratio (φ) is approximately 1.618. It's defined as (1 + √5) / 2.
\n" ); document.write( "* The dimensions are in the ratio 1:φ, meaning if the width is 'w', the length is 'wφ'.\r
\n" ); document.write( "\n" ); document.write( "**2. Use the Area Information**\r
\n" ); document.write( "\n" ); document.write( "* Area = Length * Width = 724 m²
\n" ); document.write( "* Let width = w. Then length = wφ.
\n" ); document.write( "* w * wφ = 724
\n" ); document.write( "* w²φ = 724
\n" ); document.write( "* w² = 724 / φ
\n" ); document.write( "* w = √(724 / φ)\r
\n" ); document.write( "\n" ); document.write( "**3. Calculate the Width (w)**\r
\n" ); document.write( "\n" ); document.write( "* w = √(724 / 1.618)
\n" ); document.write( "* w = √(447.466)
\n" ); document.write( "* w ≈ 21.153 m\r
\n" ); document.write( "\n" ); document.write( "**4. Calculate the Length (l)**\r
\n" ); document.write( "\n" ); document.write( "* Length (l) = wφ = 21.153 * 1.618
\n" ); document.write( "* l ≈ 34.235 m\r
\n" ); document.write( "\n" ); document.write( "**5. Use the Length Information to Find r**\r
\n" ); document.write( "\n" ); document.write( "* Length (l) = 2r - 1
\n" ); document.write( "* 34.235 = 2r - 1
\n" ); document.write( "* 35.235 = 2r
\n" ); document.write( "* r = 35.235 / 2
\n" ); document.write( "* r ≈ 17.6175\r
\n" ); document.write( "\n" ); document.write( "**6. Calculate the Perimeter (P)**\r
\n" ); document.write( "\n" ); document.write( "* Perimeter (P) = 2(Length + Width)
\n" ); document.write( "* P = 2(34.235 + 21.153)
\n" ); document.write( "* P = 2(55.388)
\n" ); document.write( "* P ≈ 110.776 m\r
\n" ); document.write( "\n" ); document.write( "**Results**\r
\n" ); document.write( "\n" ); document.write( "* **r ≈ 17.6175**
\n" ); document.write( "* **Width (w) ≈ 21.153 m**
\n" ); document.write( "* **Perimeter (P) ≈ 110.776 m**
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