A right circular cone and a sphere have equal volumes. If the cone has radius x and height 2x, what is the radius of the sphere ?

Options

A. x
B. `(x)/(root(3)(2))`
C. `root(3)(2)`
D. `(1)/(root(3)(2))`

Explanation

To solve the problem, we need to find the radius of the sphere given that the volumes of a right circular cone and a sphere are equal. The cone has a radius of \( x \) and a height of \( 2x \).

### Step-by-Step Solution:

1. **Write the formula for the volume of the cone**:
The volume \( V \) of a right circular cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height. For our cone, the radius is \( x \) and the height is \( 2x \).

2. **Substitute the values into the cone's volume formula**:
Substituting \( r = x \) and \( h = 2x \) into the volume formula:
\[ V_{cone} = \frac{1}{3} \pi x^2 (2x) = \frac{2}{3} \pi x^3 \]

3. **Write the formula for the volume of the sphere**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

4. **Set the volumes of the cone and sphere equal**:
Since the volumes of the cone and the sphere are equal, we can set the two volume equations equal to each other:
\[ \frac{2}{3} \pi x^3 = \frac{4}{3} \pi r^3 \]

5. **Cancel out common factors**:
We can cancel \( \frac{1}{3} \pi \) from both sides:
\[ 2x^3 = 4r^3 \]

6. **Rearrange the equation to solve for \( r^3 \)**:
Dividing both sides by 4 gives:
\[ r^3 = \frac{2}{4} x^3 = \frac{1}{2} x^3 \]

7. **Take the cube root to find \( r \)**:
To find \( r \), we take the cube root of both sides:
\[ r = \sqrt[3]{\frac{1}{2} x^3} = x \sqrt[3]{\frac{1}{2}} = x \frac{1}{\sqrt[3]{2}} \]

### Final Answer:
The radius of the sphere is:
\[ r = \frac{x}{\sqrt[3]{2}} \]