If the volume of a cube is equal to the volume of a sphere, what is the ratio of the edge of the cube to the radius of the sphere ?

Options

A. `1.61`
B. `2.05`
C. `2.33`
D. `2.45`

Explanation

To find the ratio of the edge of the cube to the radius of the sphere when their volumes are equal, we can follow these steps:

### Step 1: Define the variables
Let:
- \( E \) = edge length of the cube
- \( R \) = radius of the sphere

### Step 2: Write the formula for the volume of the cube
The volume \( V \) of a cube is given by the formula:
\[ V_{\text{cube}} = E^3 \]

### Step 3: Write the formula for the volume of the sphere
The volume \( V \) of a sphere is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi R^3 \]

### Step 4: Set the volumes equal to each other
According to the problem, the volume of the cube is equal to the volume of the sphere:
\[ E^3 = \frac{4}{3} \pi R^3 \]

### Step 5: Rearrange the equation to find the ratio \( \frac{E}{R} \)
To find the ratio \( \frac{E}{R} \), we can rearrange the equation:
\[ \frac{E^3}{R^3} = \frac{4}{3} \pi \]
Taking the cube root of both sides gives:
\[ \frac{E}{R} = \left(\frac{4}{3} \pi\right)^{\frac{1}{3}} \]

### Step 6: Calculate the numerical value
Now, we can substitute the approximate value of \( \pi \approx 3.14 \):
\[ \frac{E}{R} = \left(\frac{4}{3} \times 3.14\right)^{\frac{1}{3}} \]
Calculating \( \frac{4}{3} \times 3.14 \):
\[ \frac{4 \times 3.14}{3} \approx \frac{12.56}{3} \approx 4.1867 \]
Now, taking the cube root:
\[ \frac{E}{R} \approx (4.1867)^{\frac{1}{3}} \approx 1.61 \]

### Conclusion
Thus, the ratio of the edge of the cube to the radius of the sphere is approximately:
\[ \frac{E}{R} \approx 1.61 \]

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