QQuestion
Distance between two pillars is 150 m, height of one pillar is three times the other pillar. From the mid-point of the line joining their foot the angles of elevation of the top of the two pillars are complementary. Find the height of the smaller pillar
✓Answer
Let the height of the smaller pillar be h meters
Then, the height of the taller pillar is 3h meters
Let M be the midpoint of the line joining the feet of the two pillars
Since M is the midpoint, the horizontal distance from M to each pillar is:
AM = CM = \frac{150}{2} = 75 meters
Let the angles of elevation from M to the tops of the two pillars be θ and 90° – θ, since they are complementary.
In Δ MCD,
tan θ = \frac{h}{75}
In Δ AMB,
tan (90° – θ) = \frac{3h}{75}
⇒ cot θ = \frac{3h}{75}
⇒ \frac{1}{\tan θ} = \frac{3h}{75}
⇒ 75 = 3h tan θ
⇒ 25 = h tan θ
⇒ 25 = h \frac{h}{75}
⇒ h² = 25 × 75 = 1875
⇒ h = 25 √3 m
∴ Height of the smaller pillar = 25 √3 m
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