From the top of a tower, the angles of depression of two objects P and Q (situated on the ground on the same side of the tower) separated at a distance of 100 ( 3 - 3 ) m are 45° and 60 ° respectively. The height of the tower is-

Answer: Option C

Solution

Answer: Option C

Solution:

$$\begin{array}{rl}& \text{Let, }OP=a\\ & \text{tan }60^{\circ }=\frac{H}{a}\\ & \Rightarrow H=\sqrt{3}a\\ & \Rightarrow \frac{H}{\sqrt{3}}=a.....(i)\end{array}$$
$$tan45^{\circ }=1$$   $$=\frac{H}{a+100\left(3-\sqrt{3}\right)}$$
$$\Rightarrow a+100\left(3-\sqrt{3}\right)=H$$
From (i) $$\frac{H}{\sqrt{3}}+$$   $$100\left(3-\sqrt{3}\right)$$   = H
$$\begin{array}{rl}& \Rightarrow H+300\sqrt{3}-300=\sqrt{3}H\\ & \Rightarrow 300\sqrt{3}-300=\sqrt{3}H-H\\ & \Rightarrow \left(\sqrt{3}-1\right)H=300\left(\sqrt{3}-1\right)\\ & \Rightarrow H=300\text{ m}\end{array}$$