Trigonometry & Inverse Trigonometry
Heights And Distances
nta_abhyas_2025
Grade 11
Question:
Two men on the opposite sides of a tower measure the angles of elevation of the top of the tower as $45°$ and $30°$ respectively. If the height of the tower is $40$ m, then the distance between the men is
40 m
40\sqrt{3} m
68.28 m
109.28 m
Step-by-Step Solution
Key Concept: Use tangent ratios in right triangles formed by the height and horizontal distances from observation points.
From triangle $AO_1A_1$, $\tan 45° = \frac{h}{x}$, so $x = 40$ m. From triangle $AO_2AB$, $\tan 30° = \frac{h}{y}$, so $y = 40\sqrt{3}$ m. The distance between the men is $x + y = 40 + 40\sqrt{3} = 109.28$ m.
Correct Answer: 109.28