Question 1198555
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