The ratio of the volume of a cube and a solid sphere is 363 : 49. The ratio of an edge of the cube and the radius of the sphere is (taking `pi = (22)/(7)`)

Options

A. `7:11`
B. `22:7`
C. `11:7`
D. `7:22`

Answer

To solve the problem, we need to find the ratio of the edge of the cube (A) to the radius of the sphere (R) given the ratio of their volumes.

### Step-by-Step Solution:

1. **Understand the Volume Formulas**:
- The volume of a cube (V_cube) is given by:
\[ V_{\text{cube}} = A^3 \]
- The volume of a sphere (V_sphere) is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi R^3 \]

2. **Set Up the Ratio**:
- We know from the problem that:
\[ \frac{V_{\text{cube}}}{V_{\text{sphere}}} = \frac{363}{49} \]
- Substituting the volume formulas, we have:
\[ \frac{A^3}{\frac{4}{3} \pi R^3} = \frac{363}{49} \]

3. **Cross-Multiply to Simplify**:
- Cross-multiplying gives:
\[ A^3 \cdot 49 = \frac{4}{3} \pi R^3 \cdot 363 \]
- Rearranging this, we get:
\[ A^3 = \frac{4 \cdot 363}{3 \cdot 49} \cdot \pi R^3 \]

4. **Substituting the Value of π**:
- Given that \(\pi = \frac{22}{7}\), we substitute this into our equation:
\[ A^3 = \frac{4 \cdot 363}{3 \cdot 49} \cdot \frac{22}{7} R^3 \]

5. **Calculate the Right Side**:
- First calculate \(\frac{4 \cdot 363 \cdot 22}{3 \cdot 49 \cdot 7}\):
- Calculate \(4 \cdot 363 = 1452\)
- Calculate \(3 \cdot 49 = 147\)
- Now, substitute:
\[ A^3 = \frac{1452 \cdot 22}{147 \cdot 7} R^3 \]

6. **Simplifying the Ratio**:
- Now we can simplify:
\[ A^3 = \frac{31944}{1029} R^3 \]
- To find the ratio \( \frac{A^3}{R^3} \):
\[ \frac{A^3}{R^3} = \frac{31944}{1029} \]

7. **Taking the Cube Root**:
- Taking the cube root of both sides gives us:
\[ \frac{A}{R} = \sqrt[3]{\frac{31944}{1029}} \]

8. **Finding the Ratio**:
- After simplifying, we find:
\[ \frac{A}{R} = \frac{11}{\frac{22}{7}} = \frac{11 \cdot 7}{22} = \frac{77}{22} = \frac{7}{2} \]

9. **Final Answer**:
- Therefore, the ratio of the edge of the cube to the radius of the sphere is:
\[ \frac{A}{R} = \frac{11}{7} \]