<p>If \( x + y + z = 12 \), find \( x^3 + y^3 + z^3 \) given \( x = 3 \), \( y = 4 \), \( z = 5 \) (using AM-GM with \( x/3 = y/4 = z/5 \)).</p>
Step-by-Step Solution
Key Concept: When x/3 = y/4 = z/5 = k (common ratio), we can express x, y, z in terms of k and use the constraint x + y + z = 12 to find k, then compute x³ + y³ + z³ directly.
<p><strong>Step 1:</strong> Use the proportional condition x/3 = y/4 = z/5 = k (let k be the common ratio)</p><p>Then: x = 3k, y = 4k, z = 5k</p><p><strong>Step 2:</strong> Apply the constraint x + y + z = 12:</p><p>3k + 4k + 5k = 12</p><p>12k = 12</p><p>k = 1</p><p><strong>Step 3:</strong> Therefore x = 3, y = 4, z = 5</p><p><strong>Step 4:</strong> Calculate x³ + y³ + z³:</p><p>x³ + y³ + z³ = 3³ + 4³ + 5³ = 27 + 64 + 125 = 216</p><p>∴ Answer: B (216)</p>
Correct Answer: B