When the sun's angle of depression changes from $$30^\circ$$ to $$60^\circ$$. the length of the shadow of a tower decreases by 70 m. What is the height of the tower?

Solution

Let the initial distance of the shadow from the foot of the tower be x m
Then, New distance will be (x-70) m 

$$tan 30^\circ = \dfrac{h}{x}$$

=> $$\dfrac{1}{\sqrt{3}} = \dfrac{h}{x}$$

=> $$x = h\sqrt{3}$$
$$tan 60^\circ = \dfrac{h}{x-70}$$

=> $$\sqrt{3} = \dfrac{h}{x-70}$$

=> $$\sqrt{3} = \dfrac{h}{h\sqrt{3}-70}$$

=> $$h = \sqrt{3}(h\sqrt{3}-70)$$
=> $$h = 3h - 70\sqrt{3}$$
=> $$2h = 70\sqrt{3}$$
=> $$h = 35\sqrt{3} = 35\times1.73 = 60.55 m$$
Therefore, The height of the tower = 60.55 m