<p>A plane passing through \((1, 1, 1)\) cuts positive direction of co-ordinate axes at \(A\), \(B\) and \(C\), then the minimum value of the volume of tetrahedron \(OABC\)</p>
Step-by-Step Solution
Key Concept: Use the intercept form of a plane equation (x/a + y/b + z/c = 1) and apply AM-GM inequality to minimize the tetrahedron volume, which equals (1/6)abc.
Step 1: The plane cuts the positive coordinate axes at A(a,0,0), B(0,b,0), C(0,0,c). Its equation is: x/a + y/b + z/c = 1 Step 2: Since the plane passes through (1,1,1): 1/a + 1/b + 1/c = 1 Step 3: Volume of tetrahedron OABC = (1/6)abc Step 4: By AM-GM inequality: (1/a + 1/b + 1/c)/3 ≥ ∛(1/abc) Step 5: Since 1/a + 1/b + 1/c = 1: 1/3 ≥ ∛(1/abc) ⟹ abc ≥ 27 Step 6: Equality holds when 1/a = 1/b = 1/c = 1/3, so a = b = c = 3 Step 7: Minimum volume = (1/6) × 27 = 27/6 = 9/2 ∴ Answer: 9/2
Correct Answer: 9/2