Question

If the angles of depression of the upper and lower ends of a lamp post from the top of a hill of height h metres are α and β respectively, then show that the height of the lamp post (in metres) is \(\frac{h\,\text{sin (β - α)}}{\text{cos}\,\alpha\,\text{sin}\,\beta}\).

Answer 1

Let AB be the hill of height h metres and PQ be the lamp post of height \(x\) metres. 

The AB = h m, PQ = \(x\) m. 

Let BQ = y metres 

Given ∠TAP = α and ∠TAQ = β 

Draw α line PC || BQ || TA, such that C lies on AB. 

Now ∠APC = ∠TAP = α (AT || PC, alt. ∠s) 

∠AQB = ∠TAQ = β (AT || BQ, alt. ∠s) 

Also, PC || BQ, CB = PQ = \(x\) m and CP = BQ = y m. 

∴ In rt. ΔACP, tan α = \(\frac{AC}{CP}\) = \(\frac{AB - BC}{CP}\) = \(\frac{h-x}{y}\) ⇒ y = (h – x) cot α

In rt.  ΔABQ, tan β = \(\frac{AB}{BQ}\) = \(\frac{h}{y}\) ⇒ y = h cot β

∴ From (i) and (ii) (h – \(x\)) cot α = h cot β 

⇒ h cot α – h cot β = \(x\) cot α 

⇒ h \(\bigg\{\frac{\text{cos}\,\alpha}{\text{sin}\,\alpha}-\frac{\text{cos}\,\beta}{\text{sin}\,\beta}\bigg\}\) = \(x\frac{\text{cos}\,\alpha}{\text{sin}\,\alpha}\)

⇒ h \(\bigg\{\frac{\text{cos}\,\alpha\,\text{sin}\,\beta-\text{cos}\,\beta\,\text{sin}\,\alpha}{\text{sin}\,\alpha\,\text{sin}\,\beta}\bigg\}\) = \(x\frac{\text{cos}\,\alpha}{\text{sin}\,\alpha}\)

⇒ h \(\bigg\{\frac{\text{sin}\,(\beta-\alpha)}{\text{sin}\,\alpha\,\text{sin}\,\beta}\bigg\}\) = \(x\frac{\text{cos}\,\alpha}{\text{sin}\,\alpha}\) ⇒ \(x\) = \(\frac{h\,\text{sin}\,(\beta-\alpha)}{\text{cos}\,\alpha\,\text{sin}\,\beta}\).