Trigonometry & Inverse Trigonometry
Angle of Elevation and Depression
Grade 11
Question:
<p>Two poles standing on a horizontal ground are of heights 5 m and 10 m, respectively. The line joining their tops makes an angle of 15° with ground. Then, the distance (in m) between the poles is</p>
<p>(a) \(5(\sqrt{3} + 1)\)</p>
<p>(b) \(\frac{5}{2}(2 + \sqrt{3})\)</p>
<p>(c) \(10(\sqrt{3} - 1)\)</p>
<p>(d) \(5(2 + \sqrt{3})\)</p>
Step-by-Step Solution
Key Concept: Use the angle of elevation/depression concept and trigonometric ratios to find the distance between poles using the height difference and given angle.
<p><strong>Given:</strong> Heights of two poles are 5 m and 10 m.</p><p>The line joining their tops makes an angle of 15° with the ground.</p><p><strong>Solution:</strong> Let the distance between the poles be $d$ m.</p><p>The difference in heights = 10 - 5 = 5 m.</p><p>From the geometry, $\tan 15° = \frac{5}{d}$</p><p>We know $\tan 15° = 2 - \sqrt{3}$</p><p>Therefore, $d = \frac{5}{2 - \sqrt{3}} = \frac{5(2 + \sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})} = \frac{5(2 + \sqrt{3})}{4 - 3} = 5(2 + \sqrt{3})$</p><p>∴ Answer is (d).</p>
Correct Answer: d