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
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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