Question

What should be the height of a flag where a 20 ft long ladder reaches 20 ft below the flag. The angle of elevation of the top of the flag at the foot of the ladder is 60°. 

(a) 20 ft 

(b) 30 ft 

(c) 40 ft 

(d) 20√2 ft

Answer 1

(b) 30 ft

Let BD be the height of the flag. Given, DC = 20 ft. and length of ladder AC = 20 ft. 

∠DAB = 60°. 

Let BC = h metres 

In rt. Δ ABD, tan 60° = \(\frac{BD}{AB}\)

⇒ √3 = \(\frac{h+20}{AB}\)

⇒ AB = \(\frac{h+20}{\sqrt3}\) = \(\frac{(h+20)\sqrt3}{3}\)          .....(i)

In rt. Δ ABC, AC² = AB² + BC²

⇒ 20² = \(\frac{3(h+20)^2}{9}+h^2\)                    (From (i))

⇒ 400 = \(\frac{(h+20)^2+3h^2}{3}\) ⇒ 1200 = h² + 40h + 400 + 3h² 

⇒ 4h² + 40h – 800 = 0 ⇒ h² + 10h – 200 = 0 

(h + 20) (h – 10) = 0 ⇒ h = 10             (∵ h ≠ – 20) 

∴ Height of the flag = BC + CD = 10 + 20 = 30 ft.