A man observes the angle of elevation of the top of the tower to be 45°. He walks towards it in a horizontal line through its base. On covering 20 m, the angle of elevation changes to 60°. Find the height of the tower correct to 2 significant figures.

Let tower be QR and initial position of man be P, since the initial angle of elevation is 45°, considering right angled △PQR we get,

$\Rightarrow \tan 45° = \dfrac{\text{perpendicular}}{\text{base}} \\[1em] \Rightarrow 1 = \dfrac{QR}{PQ} \\[1em] \Rightarrow PQ = QR.$

After covering 20 m let the man be at point S, so PS = 20 m and SQ = PQ - PS = PQ - 20 = QR - 20.

Now considering right angled △SQR we get,

$\Rightarrow \tan 60^{\circ} = \dfrac{QR}{SQ} \\[1em] \Rightarrow \sqrt3 = \dfrac{QR}{QR-20} \\[1em] \Rightarrow \sqrt3(QR - 20) = QR \\[1em] \Rightarrow QR\sqrt3 - 20\sqrt3 = QR \\[1em] \Rightarrow QR\sqrt3 - QR = 20(1.732) \\[1em] \Rightarrow QR(\sqrt3 - 1) = 34.64 \\[1em] \Rightarrow QR(1.732 - 1) = 34.64 \\[1em] \Rightarrow 0.732QR = 34.64 \\[1em] \Rightarrow QR = \dfrac{34.64}{0.732} \\[1em] \Rightarrow QR = 47.32.$

On correcting to 2 significant figures QR = 47.

Hence, the height of the tower is 47 m.