A largest posible cone is cut out from a cube of volume ` 1000 cm^(3)` . The volume of the cone is :

Options

A. `280 cm^(3) `
B. `261.9 cm^(3) `
C. `269.1 cm^(3) `
D. ` 296.1 cm^(3)`

Answer

To find the volume of the largest possible cone that can be cut out from a cube with a volume of 1000 cm³, we can follow these steps:

### Step-by-Step Solution:

1. **Find the side length of the cube:**
The volume of a cube is given by the formula:
\[ V = \text{side}^3 \]
Given that the volume \( V = 1000 \, \text{cm}^3 \), we can find the side length:
\[ \text{side}^3 = 1000 \]
Taking the cube root of both sides:
\[ \text{side} = \sqrt[3]{1000} = 10 \, \text{cm} \]

2. **Determine the dimensions of the cone:**
The largest cone that can be inscribed in the cube will have its diameter equal to the side of the cube. Therefore:
- Diameter of the cone = Side of the cube = 10 cm
- Radius of the cone \( r \) = Diameter / 2 = \( \frac{10}{2} = 5 \, \text{cm} \)
- Height of the cone \( h \) = Side of the cube = 10 cm

3. **Calculate the volume of the cone:**
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V = \frac{1}{3} \pi (5)^2 (10) \]
\[ V = \frac{1}{3} \pi (25)(10) = \frac{250}{3} \pi \]

4. **Using \( \pi \approx \frac{22}{7} \) for calculation:**
\[ V \approx \frac{250}{3} \times \frac{22}{7} = \frac{250 \times 22}{3 \times 7} = \frac{5500}{21} \approx 261.90 \, \text{cm}^3 \]

### Final Answer:
The volume of the largest possible cone that can be cut out from the cube is approximately:
\[ \boxed{261.90 \, \text{cm}^3} \]