<p>Given 5, \(5r\), \(5r^2\) are the lengths of the sides of a triangle, find the least value of \(r\).</p>
Step-by-Step Solution
Key Concept: For a valid triangle, the sum of any two sides must exceed the third side. The critical constraint is that the two smaller sides must sum to more than the largest side, which gives us the binding inequality for the minimum value of r.
<p><strong>Step 1:</strong> Let the sides be 5, 5r, 5r² where r > 0. For a valid triangle, all three triangle inequalities must hold.</p><p><strong>Step 2:</strong> The three inequalities are:<br>• 5 + 5r > 5r² ⟹ 1 + r > r²<br>• 5 + 5r² > 5r ⟹ 1 + r² > r (always true for r > 0)<br>• 5r + 5r² > 5 ⟹ r + r² > 1</p><p><strong>Step 3:</strong> From inequality (1): r² - r - 1 < 0. Using the quadratic formula:<br>r = (1 ± √5)/2. Since r > 0, we need r < (1 + √5)/2 ≈ 1.618</p><p><strong>Step 4:</strong> From inequality (3): r² + r - 1 > 0. Using the quadratic formula:<br>r = (-1 ± √5)/2. Since r > 0, we need r > (-1 + √5)/2 ≈ 0.618</p><p><strong>Step 5:</strong> The binding lower constraint is r > (√5 - 1)/2 = φ - 1, where φ is the golden ratio.</p><p>∴ Minimum value of r = (√5 - 1)/2 ≈ <strong>0.6180</strong></p>
Correct Answer: 0.6180