A solid spherical ball is prepared by melting a cone and a cylinder having the same height and same base diameter equal to 2r. Find the radius of the sphere.

Options

A. `2^(1//3)r`
B. `2^(1//3)r^(2)`
C. `4^(1//3)r`
D. `18^(1//3)r`

Answer

To find the radius of the sphere formed by melting a cone and a cylinder with the same height and base diameter, we can follow these steps:

### Step 1: Define the dimensions of the cone and cylinder
- Given that the diameter of both the cone and the cylinder is \(2r\), the radius \(R\) of both shapes is:
\[ R = \frac{2r}{2} = r \]
- The height \(h\) of both the cone and the cylinder is given as \(2r\).

### Step 2: Calculate the volume of the cone
- The formula for the volume \(V\) of a cone is:
\[ V = \frac{1}{3} \pi R^2 h \]
- Substituting the values of \(R\) and \(h\):
\[ V_{\text{cone}} = \frac{1}{3} \pi (r^2) (2r) = \frac{2}{3} \pi r^3 \]

### Step 3: Calculate the volume of the cylinder
- The formula for the volume \(V\) of a cylinder is:
\[ V = \pi R^2 h \]
- Substituting the values of \(R\) and \(h\):
\[ V_{\text{cylinder}} = \pi (r^2) (2r) = 2 \pi r^3 \]

### Step 4: Calculate the total volume of the cone and cylinder
- The total volume \(V_{\text{total}}\) is the sum of the volumes of the cone and the cylinder:
\[ V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = \frac{2}{3} \pi r^3 + 2 \pi r^3 \]
- To combine these volumes, we need a common denominator:
\[ V_{\text{total}} = \frac{2}{3} \pi r^3 + \frac{6}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

### Step 5: Set the total volume equal to the volume of the sphere
- The volume \(V\) of a sphere is given by:
\[ V = \frac{4}{3} \pi R_s^3 \]
- Setting the total volume equal to the volume of the sphere:
\[ \frac{4}{3} \pi R_s^3 = \frac{8}{3} \pi r^3 \]

### Step 6: Solve for the radius of the sphere \(R_s\)
- Cancel \(\frac{4}{3} \pi\) from both sides:
\[ R_s^3 = 2r^3 \]
- Taking the cube root of both sides:
\[ R_s = \sqrt[3]{2} r \]

### Final Answer
The radius of the sphere is:
\[ R_s = 2^{1/3} r \]