Two observers are stationed due north of a tower (of height x metre) at a distance y metre from each other. The angles of elevation of the tower observed by them are 30° and 45° respectively. Then $frac{x}{y}$ is equal to

$angle mathrm{PSQ}=45^{circ}, angle mathrm{PRQ}=30^{circ}$

From ΔPQS

$an 45^{circ}=frac{mathrm{PQ}}{mathrm{QS}}$

$Rightarrow 1=frac{x}{mathrm{QS}} Rightarrow mathrm{QS}=x$ metre

From ΔPQR,

$an 30^{circ}=frac{mathrm{PQ}}{mathrm{QR}}$

$Rightarrow frac{1}{sqrt{3}}=frac{x}{x+y}$

$Rightarrow sqrt{3} x=x+y$

$Rightarrow sqrt{3} x-x=y$

$Rightarrow x(sqrt{3}-1)=y$

$Rightarrow frac{x}{y}=frac{1}{sqrt{3}-1}=frac{sqrt{3}+1}{(sqrt{3}-1)(sqrt{3}+1)}$

$=frac{sqrt{3}+1}{3-1}=frac{sqrt{3}+1}{2}$ metre