At a particular time, when the sun's altitude is 30°, the length of the shadow of a vertical tower is 45 m. Calculate :

(i) the height of the tower,

(ii) the length of the shadow of the same tower, when the sun's altitude is :

(a) 45° (b) 60°.

(i) Let AB be the tower.

From figure,

In △ABC,

$\text{tan 30°} = \dfrac{\text{Perpendicular}}{\text{Base}} \\[1em] \Rightarrow \dfrac{1}{\sqrt{3}} = \dfrac{AB}{BC} \\[1em] \Rightarrow \dfrac{h}{45} = \dfrac{1}{\sqrt{3}} \\[1em] \Rightarrow h = \dfrac{45}{\sqrt{3}} \\[1em] \Rightarrow h = \dfrac{45}{1.732} \\[1em] \Rightarrow h = 25.98 \text{ meters}.$

Hence, the height of tower = 25.98 meters.

(ii) When sun's altitude is

(a) 45°

$\Rightarrow \text{tan 45°} = \dfrac{AB}{BC} \\[1em] \Rightarrow 1 = \dfrac{h}{BC} \\[1em] \Rightarrow BC = h = 25.98 \text{ meters}.$

Hence, length of shadow of tower when the sun's altitude is 45° = 25.98 meters.

(b) 60°

$\Rightarrow \text{tan 60°} = \dfrac{AB}{BC} \\[1em] \Rightarrow \sqrt{3} = \dfrac{h}{BC} \\[1em] \Rightarrow BC = \dfrac{h}{\sqrt{3}} \\[1em] \Rightarrow BC = \dfrac{25.98}{1.732} \\[1em] \Rightarrow BC = 15\text{ meters}.$

Hence, length of shadow of tower when the sun's altitude is 60° = 15 meters.