A plane passing through (1, 1, 1) cuts positive direction of co-ordinate axes at A, B and C if V be the volume of tetrahedron OABC, then the minimum value of V is __________

Answer

To find the minimum volume \( V \) of the tetrahedron \( OABC \) where the plane passes through the point \( (1, 1, 1) \) and intersects the coordinate axes at points \( A \), \( B \), and \( C \), we can follow these steps:

### Step 1: Define the coordinates of points A, B, and C
Let the coordinates of points \( A \), \( B \), and \( C \) be:
- \( A(a, 0, 0) \) on the x-axis
- \( B(0, b, 0) \) on the y-axis
- \( C(0, 0, c) \) on the z-axis

### Step 2: Write the equation of the plane
The equation of the plane that intersects the axes at points \( A \), \( B \), and \( C \) can be expressed as:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]

### Step 3: Substitute the point (1, 1, 1)
Since the plane passes through the point \( (1, 1, 1) \), we substitute \( x = 1 \), \( y = 1 \), and \( z = 1 \) into the plane equation:
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1 \]

### Step 4: Express the volume of the tetrahedron
The volume \( V \) of the tetrahedron \( OABC \) is given by the formula:
\[ V = \frac{1}{6} \cdot a \cdot b \cdot c \]

### Step 5: Use the AM-GM inequality
To find the minimum value of \( V \), we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality. According to AM-GM:
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
From the equation \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1 \), we can express \( abc \) in terms of \( a, b, c \):
\[ abc \geq 27 \quad \text{(from AM-GM)} \]

### Step 6: Substitute into the volume formula
Substituting the minimum value of \( abc \) into the volume formula:
\[ V = \frac{1}{6} \cdot abc \geq \frac{1}{6} \cdot 27 = \frac{27}{6} = 4.5 \]

### Conclusion
Thus, the minimum volume \( V \) of the tetrahedron \( OABC \) is:
\[ \boxed{4.5} \]