A cylinder having base radius equal to its height, right circular cone whose radius is equal to its height and a hemisphere are given . All of them have the same volumes. What will be the ratio of their respective radius?

Options

A. `root3(3) : 1: root3(2)`
B. `1: root3(2): root3(2)`
C. `root3(2) : root3(6): root3(3)`
D. `root3(4): root3(6): root2(3)`

Answer

To solve the problem, we need to find the ratio of the radii of a cylinder, a cone, and a hemisphere, given that they all have the same volume and that the height of the cylinder and cone is equal to their respective radii.

### Step-by-Step Solution:

1. **Define the Variables:**
Let the radius of the cylinder be \( r_1 \), the radius of the cone be \( r_2 \), and the radius of the hemisphere be \( r_3 \). According to the problem, the height of the cylinder \( h \) is equal to its radius \( r_1 \), and the height of the cone \( h \) is equal to its radius \( r_2 \).

2. **Write the Volume Formulas:**
- Volume of the cylinder:
\[ V_{\text{cylinder}} = \pi r_1^2 h = \pi r_1^3 \]
- Volume of the cone:
\[ V_{\text{cone}} = \frac{1}{3} \pi r_2^2 h = \frac{1}{3} \pi r_2^3 \]
- Volume of the hemisphere:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r_3^3 \]

3. **Set the Volumes Equal:**
Since all volumes are equal, we can set them equal to each other:
\[ \pi r_1^3 = \frac{1}{3} \pi r_2^3 = \frac{2}{3} \pi r_3^3 \]

4. **Cancel \(\pi\) from the Equations:**
Dividing through by \(\pi\):
\[ r_1^3 = \frac{1}{3} r_2^3 \]
\[ r_1^3 = \frac{2}{3} r_3^3 \]

5. **Express \(r_2\) and \(r_3\) in Terms of \(r_1\):**
From \(r_1^3 = \frac{1}{3} r_2^3\):
\[ r_2^3 = 3 r_1^3 \implies r_2 = r_1 \cdot 3^{1/3} \]
From \(r_1^3 = \frac{2}{3} r_3^3\):
\[ r_3^3 = \frac{3}{2} r_1^3 \implies r_3 = r_1 \cdot \left(\frac{3}{2}\right)^{1/3} \]

6. **Find the Ratio \(r_1 : r_2 : r_3\):**
Now we can express the ratio:
\[ r_1 : r_2 : r_3 = r_1 : (r_1 \cdot 3^{1/3}) : \left(r_1 \cdot \left(\frac{3}{2}\right)^{1/3}\right) \]
Simplifying this gives:
\[ 1 : 3^{1/3} : \left(\frac{3}{2}\right)^{1/3} \]

7. **Final Ratio:**
To express the ratio in a more standard form, we can multiply through by \(2^{1/3}\):
\[ 2^{1/3} : 2^{1/3} \cdot 3^{1/3} : 3^{1/3} \]
This simplifies to:
\[ 2^{1/3} : 3^{1/3} : 2^{1/3} \cdot 3^{1/3} \]

### Conclusion:
The final ratio of the respective radii \( r_1 : r_2 : r_3 \) is:
\[ 1 : 3^{1/3} : \left(\frac{3}{2}\right)^{1/3} \]