A building is 8 m high. From its top, the angle of elevation of the top of a tower is 60°, and the angle of depression of its foot is 30°. Find the height of the tower and the distance between them. (Use √3 = 1.732)
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found Given: building AB = 8 m; from A, elevation of the tower top D = 60°, depression of its foot C = 30°. To find: CD and BC.
2 · Draw the figure Verticals AB (short) and CD (tall) on the ground BC. Horizontal AE from A to the tower. Mark 60° above AE to D and 30° below AE to C.
3 · Write the formula or theorem Angle of depression = angle of elevation (alternate angles). tan θ = opposite/adjacent.
4 · Find the distance In ΔABC: tan 30° = AB/BC, so 1/√3 = 8/BC and BC = 8√3 m. So AE = 8√3 m.
5 · Find the part above eye level In ΔAED: tan 60° = DE/AE, so √3 = DE/(8√3) and DE = 24 m.
6 · Add the heights CE = AB = 8 m, so CD = CE + DE = 8 + 24 = 32 m.
7 · Use the value of √3 BC = 8 × 1.732 = 13.856 ≈ 13.86 m.
8 · Check and box the answer Tower = 32 m; distance between them ≈ 13.86 m.
Scoring tip: Write AE = BC and CE = AB in words. Examiners give a mark for spotting the rectangle ABCE.
