|
Question 1198563: 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 CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
Certainly, let'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 'h' 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: 'h'
* 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.**
|
|
|
| |
