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ABHYAS PREP: RACES, DISCOUNTS, STOCKS AND HEIGHTS · HEIGHT & DISTANCE

Heights and distances with 30°, 45° and 60°

Draw the right triangle. tan θ = opposite/adjacent = height/horizontal distance; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. An angle of depression from the top equals the angle of elevation from below.

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Worked examples

Example 1 · level 1

From the top of a watch tower on the coast 72 m above the water, Fatima sees a fishing boat at an angle of depression of 30°. How far is the boat from the foot of the watch tower on the coast?

  1. 144 m
  2. 24√3 m
  3. 72√3 m
  4. 48√3 m
Show solution
  1. Angle of elevation of the top from the boat = 30° (alternate angles).
  2. tan 30° = 72/distance ⇒ distance = 72 × √3 = 72√3 m.

Answer: 72√3 m

Example 2 · level 2

Saanvi looks at the top of a flagpole in Ranchi and finds the angle of elevation is 30°. After walking 30 m straight towards it, the angle becomes 45°. How tall is it?

  1. 15(√3 + 1) m
  2. 15(√3 − 1) m
  3. 30 m
  4. 10(3 − √3) m
Show solution
  1. Far point: distance = h cot 30° = √3·h. Near point: distance = h cot 45° = 1·h.
  2. √3·h − 1·h = 30 ⇒ h = 15(√3 + 1) m.
  3. Distance from the near point = h × 1 = 15(√3 + 1) m.

Answer: 15(√3 + 1) m

Example 3 · level 3

From the top of a 54 m building in Patna, Aarav sees the top of a flagpole across the road at an angle of elevation of 30° and its foot at an angle of depression of 45°. How tall is the flagpole?

  1. 18(3 + √3) m
  2. 54(√3 + 1) m
  3. 18(3 − √3) m
  4. 18√3 m
Show solution
  1. Gap between them = 54 × cot 45° = 54 × 1 = 54 m.
  2. Part above the roof = gap × tan 30° = 18√3 m.
  3. Height = 54 + 18√3 = 18(3 + √3) m.

Answer: 18(3 + √3) m

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