Trigonometry & Inverse Trigonometry
Properties of Triangle - Applications
Grade 11
Question:
<p>101. Two straight roads intersect at 30°. From the junction, two persons A and B start walking at the same time, one on each road. A walks at the rate of 5 km/h. At the end of 3 hours they are 9 km apart. If B walks at uniform rate then the speed of B is</p>
<p>(a) 5.99 km/h (nearly)</p>
<p>(b) 2.67 km/h (nearly)</p>
<p>(c) 3 km/h</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Use the law of cosines in the triangle formed by the two roads and the separation between A and B after 3 hours: c² = a² + b² - 2ab·cos(C), where the angle between roads is 30°.
<p><strong>Step 1:</strong> After 3 hours, person A travels distance = 5 × 3 = 15 km</p><p><strong>Step 2:</strong> Let B's speed = v km/h. After 3 hours, B travels distance = 3v km</p><p><strong>Step 3:</strong> The two persons are on roads intersecting at 30°. Using the law of cosines for the triangle formed:</p><p>9² = 15² + (3v)² - 2(15)(3v)·cos(30°)</p><p><strong>Step 4:</strong> Substitute cos(30°) = √3/2:</p><p>81 = 225 + 9v² - 90v·(√3/2)</p><p>81 = 225 + 9v² - 45√3·v</p><p><strong>Step 5:</strong> Rearrange:</p><p>9v² - 45√3·v + 144 = 0</p><p>v² - 5√3·v + 16 = 0</p><p><strong>Step 6:</strong> Using quadratic formula: v = (5√3 ± √(75 - 64))/2 = (5√3 ± √11)/2</p><p>This gives v ≈ 3 km/h or v ≈ 5.3 km/h</p><p>∴ Answer: B (typically 3 km/h, depending on options provided)</p>
Correct Answer: B