<p>Let the equation of the plane be \(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\). The volume of tetrahedron \(OABC\) is \(V = \frac{1}{6}(abc)\). Given that \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1\), find the minimum value of \(V\).</p>
Step-by-Step Solution
Key Concept: Use the constraint ∑(1/a) = 1 with AM-GM inequality on the reciprocals to minimize abc, recognizing that the volume V = (1/6)abc is minimized when abc is minimized under this linear constraint.
Step 1: Set up the optimization problem. We need to minimize V = (1/6)abc subject to the constraint 1/a + 1/b + 1/c = 1, where a, b, c > 0. Step 2: Apply AM-GM inequality to the reciprocals. By AM-GM inequality: ∑(1/a)/3 ≥ ∛(1/a · 1/b · 1/c) (1/3)(1/a + 1/b + 1/c) ≥ ∛(1/(abc)) (1/3)(1) ≥ ∛(1/(abc)) Step 3: Cube both sides to solve for abc: 1/27 ≥ 1/(abc) abc ≥ 27 Step 4: Equality holds in AM-GM when 1/a = 1/b = 1/c, which gives a = b = c. Substituting into the constraint: 3(1/a) = 1 → a = 3 Therefore a = b = c = 3, and abc = 27. Step 5: Calculate the minimum volume: V_min = (1/6) × 27 = 27/6 = 4.5 ∴ Answer: 4.50
Correct Answer: 4.50