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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.
Answer by onyulee(41)   (Show Source):
You can put this solution on YOUR website! Certainly, let's find the total area and volume of the cone.
**1. Find the Radius of the Cone's Base**
* The arc length of the sector becomes the circumference of the cone's base.
* Arc length = (central angle / 360) * 2 * π * radius
* Arc length = (120/360) * 2 * π * 20 = (1/3) * 40π = (40/3)π inches
* Circumference of the base = 2 * π * cone_radius
* cone_radius = (40/3)π / (2 * π) = 20/3 inches
**2. Find the Slant Height of the Cone**
* The slant height of the cone is equal to the radius of the sector, which is 20 inches.
**3. Find the Height of the Cone**
* Using the Pythagorean theorem:
* height² = slant height² - radius²
* height² = 20² - (20/3)²
* height² = 400 - 400/9
* height² = 3600/9 - 400/9
* height² = 3200/9
* height = √(3200/9) = (40√2)/3 inches
**4. Find the Total Surface Area of the Cone**
* Total Surface Area = π * radius * (radius + slant height)
* Total Surface Area = π * (20/3) * (20/3 + 20)
* Total Surface Area = π * (20/3) * (80/3)
* Total Surface Area = (1600/9)π square inches
**5. Find the Volume of the Cone**
* Volume = (1/3) * π * radius² * height
* Volume = (1/3) * π * (20/3)² * (40√2)/3
* Volume = (1/3) * π * (400/9) * (40√2)/3
* Volume = (16000√2/81)π cubic inches
**6. Express the Volume in the Given Form**
* V = (16000√2/81)π
* V = (16000/81) * π * √2
* Comparing with V = βπ / σ √γ:
* β = 16000
* σ = 81
* γ = 2
**7. Find the Smallest Sum of β, γ, and σ**
* Sum = β + γ + σ = 16000 + 2 + 81 = 16083
**Therefore:**
* Total Surface Area of the Cone: (1600/9)π square inches
* Volume of the Cone: (16000√2/81)π cubic inches
* Smallest sum of β, γ, and σ: 16083
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