(c) \(\frac{a\sqrt3(\sqrt3+1)}{2}\)
Let AB be the tower of height h metres (say).
Given,
building PQ = a metres. Draw PR || BQ such that R lies on AB.
⇒ PR = BQ
Given,
∠APR = 30° and ∠AQB = 45°
RB = PQ = a
⇒ AR = (h – a)
∴ In rt. Δ ARP,
tan 30° = \(\frac{AR}{RP}=\frac{h-a}{RP}\)
⇒ \(\frac{1}{\sqrt3}=\frac{h-a}{RP}\)
⇒ RP = (h-a)\(\sqrt3\) ...(i)
In rt. Δ ABQ,
tan 45° = \(\frac{AB}{BQ}=\frac{h}{BQ}\)
⇒ 1 = \(\frac{h}{BQ}\)
⇒ BQ = h ...(ii)
∴ RP = BQ, from (i) and (ii), we have,
h = (h - a)\(\sqrt3\)
⇒ h(\(\sqrt3\) - 1) = a\(\sqrt3\)
⇒ h = \(\frac{a\sqrt3}{\sqrt3-1}\times\frac{\sqrt3+1}{\sqrt3+1}\)
= \(\frac{a\sqrt3(\sqrt3+1)}{2}\)