The greatest possible sphere is turned from a cubical block of wood. If the volume of the block removed be 35280 cu.in., the diameter of the sphere `(pi = 22//.7)` will be

Options

A. 33 in.
B. 27 in.
C. 39 in.
D. None of these

Answer

To find the diameter of the greatest possible sphere that can be turned from a cubical block of wood with a given volume of 35280 cubic inches, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We are given the volume of wood removed (which is the volume of the sphere) as 35280 cubic inches. We need to find the diameter of the sphere.

2. **Volume of the Cube**:
Let the side length of the cube be \( x \). The volume of the cube is given by:
\[ V_{\text{cube}} = x^3 \]

3. **Volume of the Sphere**:
The volume of the sphere is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Since the diameter \( d \) of the sphere is equal to the side length \( x \) of the cube, we have \( r = \frac{x}{2} \). Thus, the volume of the sphere can be rewritten as:
\[ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{x}{2}\right)^3 = \frac{4}{3} \pi \frac{x^3}{8} = \frac{\pi x^3}{6} \]

4. **Setting Up the Equation**:
The volume of the wood removed (the volume of the sphere) is equal to the volume of the cube minus the volume of the sphere:
\[ V_{\text{removed}} = V_{\text{cube}} - V_{\text{sphere}} \]
Substituting the volumes we have:
\[ 35280 = x^3 - \frac{\pi x^3}{6} \]

5. **Simplifying the Equation**:
Factor out \( x^3 \):
\[ 35280 = x^3 \left(1 - \frac{\pi}{6}\right) \]
Substitute \( \pi = \frac{22}{7} \):
\[ 1 - \frac{22}{7 \cdot 6} = 1 - \frac{22}{42} = 1 - \frac{11}{21} = \frac{10}{21} \]
Therefore, we can rewrite the equation as:
\[ 35280 = x^3 \cdot \frac{10}{21} \]

6. **Solving for \( x^3 \)**:
Rearranging gives:
\[ x^3 = 35280 \cdot \frac{21}{10} = 35280 \cdot 2.1 = 74088 \]

7. **Finding \( x \)**:
To find \( x \), we take the cube root:
\[ x = \sqrt[3]{74088} \]
Calculating the cube root gives:
\[ x = 42 \]

8. **Finding the Diameter**:
Since the diameter of the sphere is equal to the side length of the cube:
\[ \text{Diameter} = x = 42 \text{ inches} \]

### Final Answer:
The diameter of the sphere is **42 inches**.