SOLUTION: The bases of a frustum of a right circular cone are circles whose diameters are respectively 18 and 14 inches. With a slant height of 25 inches, the volume may be expressed as V =

Question 1198562: The bases of a frustum of a right circular cone are circles whose diameters are respectively 18 and 14 inches. With a slant height of 25 inches, the volume may be expressed as V = Χπ √γ in^3 where Χ and γ are integers. Find the smallest sum of Χ and γ.
Answer by onyulee(41) (Show Source): You can put this solution on YOUR website!
**1. Find the Radii**
* 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 of the Frustum**
* Let 'h' be the height of the frustum.
* We have a right triangle formed by:
* Slant height (l) = 25 inches
* Height (h)
* Difference in radii (R - r) = 9 - 7 = 2 inches
* Using the Pythagorean Theorem:
* h² = l² - (R - r)²
* h² = 25² - 2²
* h² = 625 - 4
* h² = 621
* h = √621 inches
**3. Calculate the Volume of the Frustum**
* Volume (V) = (1/3) * π * h * (R² + Rr + r²)
* 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 Given Form**
* V = (193/3) * π * √621
* V = Χπ√γ
* **Χ = 193/3**
* **γ = 621**
**5. Find the Smallest Sum of Χ and γ**
* Sum = Χ + γ = (193/3) + 621
* Sum = 193/3 + 1863/3
* Sum = 2056/3
**Therefore, the smallest sum of Χ and γ is 2056/3.**

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