document.write( "Question 1198555: A circular sector has a radius of 20 in. and a central angle of 120°. If this sector is cut out of paper and rolled so as to form the lateral surface of a right circular cone, find the total area and volume of the cone. The volume of the solid generated by this triangle may be expressed as V = βπ / σ √γ in^3 where β and σ are positive integers and γ is a prime number. Find the smallest sum of β,γ, and σ. \n" ); document.write( "
Algebra.Com's Answer #848334 by onyulee(41)\"\" \"About 
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Certainly, let's find the total area and volume of the cone.\r
\n" ); document.write( "\n" ); document.write( "**1. Find the Radius of the Cone's Base**\r
\n" ); document.write( "\n" ); document.write( "* The arc length of the sector becomes the circumference of the cone's base.
\n" ); document.write( "* Arc length = (central angle / 360) * 2 * π * radius
\n" ); document.write( "* Arc length = (120/360) * 2 * π * 20 = (1/3) * 40π = (40/3)π inches\r
\n" ); document.write( "\n" ); document.write( "* Circumference of the base = 2 * π * cone_radius
\n" ); document.write( "* cone_radius = (40/3)π / (2 * π) = 20/3 inches\r
\n" ); document.write( "\n" ); document.write( "**2. Find the Slant Height of the Cone**\r
\n" ); document.write( "\n" ); document.write( "* The slant height of the cone is equal to the radius of the sector, which is 20 inches.\r
\n" ); document.write( "\n" ); document.write( "**3. Find the Height of the Cone**\r
\n" ); document.write( "\n" ); document.write( "* Using the Pythagorean theorem:
\n" ); document.write( " * height² = slant height² - radius²
\n" ); document.write( " * height² = 20² - (20/3)²
\n" ); document.write( " * height² = 400 - 400/9
\n" ); document.write( " * height² = 3600/9 - 400/9
\n" ); document.write( " * height² = 3200/9
\n" ); document.write( " * height = √(3200/9) = (40√2)/3 inches\r
\n" ); document.write( "\n" ); document.write( "**4. Find the Total Surface Area of the Cone**\r
\n" ); document.write( "\n" ); document.write( "* Total Surface Area = π * radius * (radius + slant height)
\n" ); document.write( "* Total Surface Area = π * (20/3) * (20/3 + 20)
\n" ); document.write( "* Total Surface Area = π * (20/3) * (80/3)
\n" ); document.write( "* Total Surface Area = (1600/9)π square inches\r
\n" ); document.write( "\n" ); document.write( "**5. Find the Volume of the Cone**\r
\n" ); document.write( "\n" ); document.write( "* Volume = (1/3) * π * radius² * height
\n" ); document.write( "* Volume = (1/3) * π * (20/3)² * (40√2)/3
\n" ); document.write( "* Volume = (1/3) * π * (400/9) * (40√2)/3
\n" ); document.write( "* Volume = (16000√2/81)π cubic inches\r
\n" ); document.write( "\n" ); document.write( "**6. Express the Volume in the Given Form**\r
\n" ); document.write( "\n" ); document.write( "* V = (16000√2/81)π
\n" ); document.write( "* V = (16000/81) * π * √2 \r
\n" ); document.write( "\n" ); document.write( "* Comparing with V = βπ / σ √γ:
\n" ); document.write( " * β = 16000
\n" ); document.write( " * σ = 81
\n" ); document.write( " * γ = 2 \r
\n" ); document.write( "\n" ); document.write( "**7. Find the Smallest Sum of β, γ, and σ**\r
\n" ); document.write( "\n" ); document.write( "* Sum = β + γ + σ = 16000 + 2 + 81 = 16083\r
\n" ); document.write( "\n" ); document.write( "**Therefore:**\r
\n" ); document.write( "\n" ); document.write( "* Total Surface Area of the Cone: (1600/9)π square inches
\n" ); document.write( "* Volume of the Cone: (16000√2/81)π cubic inches
\n" ); document.write( "* Smallest sum of β, γ, and σ: 16083
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