<PRE><B>Question 1198563</B>


Certainly, let&#39;s find the volume of the frustum and determine the smallest sum of Χ and γ.

**1. Find the radii of the bases:**

* Radius of the larger base (R): 18 inches / 2 = 9 inches
* Radius of the smaller base (r): 14 inches / 2 = 7 inches

**2. Find the height (h) of the frustum:**

* We can use the Pythagorean theorem with the slant height (l = 25 inches), the radius difference (R - r = 2 inches), and the height (h):

   l² = (R - r)² + h²
   25² = 2² + h²
   h² = 625 - 4
   h² = 621
   h = √621 inches

**3. Calculate the volume (V) of the frustum:**

* The formula for the volume of a frustum of a right circular cone is:

   V = (1/3) * π * h * (R² + Rr + r²) 

* Substitute the values:

   V = (1/3) * π * √621 * (9² + 9*7 + 7²)
   V = (1/3) * π * √621 * (81 + 63 + 49)
   V = (1/3) * π * √621 * 193

**4. Express the volume in the form V = Χπ√γ**

* V = (1/3) * π * √621 * 193 
* V = (193/3) * π * √621

* Therefore: 
    * Χ = 193
    * γ = 621

**5. Find the smallest sum of Χ and γ**

* Χ + γ = 193 + 621 = 814

**Therefore, the smallest sum of Χ and γ is 814.**
**1. Find the Radii**

* **Radius of the larger base:** 18 inches / 2 = 9 inches
* **Radius of the smaller base:** 14 inches / 2 = 7 inches

**2. Find the Height of the Frustum**

* Let &#39;h&#39; be the height of the frustum.
* We can use the Pythagorean theorem with the slant height and the difference in radii to find the height.

* Consider a right triangle formed by:
    * The base: Difference in radii = 9 inches - 7 inches = 2 inches
    * The height: &#39;h&#39;
    * The hypotenuse: Slant height = 25 inches

* Using Pythagorean Theorem:
    * h² = 25² - 2²
    * h² = 625 - 4
    * h² = 621
    * h = √621 inches

**3. Find the Volume of the Frustum**

* The formula for the volume (V) of a frustum of a right circular cone is:

   V = (1/3) * π * h * (R² + Rr + r²) 

   where:
      * h is the height of the frustum 
      * R is the radius of the larger base (9 inches)
      * r is the radius of the smaller base (7 inches)

* V = (1/3) * π * √621 * (9² + 9*7 + 7²) 
* V = (1/3) * π * √621 * (81 + 63 + 49) 
* V = (1/3) * π * √621 * 193

**4. Express the Volume in the Form V = Χπ√γ**

* V = (193/3) * π * √621 

* **Therefore:**
    * Χ = 193/3 
    * γ = 621

**5. Find the Smallest Sum of Χ and γ**

* Sum = Χ + γ = (193/3) + 621 
* Sum = (193 + 1863)/3 
* Sum = 2056/3 

**Therefore, the smallest sum of Χ and γ is 2056/3.**
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