Final answer:
The volume of the largest right circular cone that can be inscribed in a sphere of radius 18 is approximately 5832π√3.
Step-by-step explanation:
To find the volume of the largest right circular cone that can be inscribed in a sphere, we can use the following steps:
- First, let's determine the diameter of the sphere, which is twice the radius. So, the diameter is 2 * 18 = 36 units.
- Next, we find the height of the cone by using the Pythagorean Theorem. The height, radius, and slant height of the cone form a right triangle. The slant height of the cone is equal to the radius of the sphere, which is 18 units. Using the Pythagorean Theorem, we can solve for the height:
- Height² + radius² = slant height²
- Height² + 18² = 36²
- Height² + 324 = 1296
- Height² = 972
- Height = √972
- Height = 18√3
Now, we can use the formula for finding the volume of a cone: V = (1/3)πr²h, where r is the radius and h is the height. Plugging in the values, we get:
- V = (1/3)π(18)²(18√3)
- V = (1/3)3.14(324)(18√3)
- V ≈ 324π(18√3)
- V ≈ 5832π√3
Therefore, the volume of the largest right circular cone that can be inscribed in a sphere of radius 18 is approximately 5832π√3.