When the sun's angle of elevation is 45°, the length of the shadow of a vertical pillar on a horizontal plane is a certain value. When the angle of elevation becomes 30°, the shadow length increases by 60 meters compared to the previous case. Find the height of the pillar.

Q.When the sun's angle of elevation is 45°, the length of the shadow of a vertical pillar on a horizontal plane is a certain value. When the angle of elevation becomes 30°, the shadow length increases by 60 meters compared to the previous case. Find the height of the pillar.

Let’s assume that when the sun’s angle of elevation is 45°, the length of the shadow is BC. When the angle of elevation is 30°, the shadow length becomes BD, and the increased shadow length is CD = 60 meters. From the right-angled triangle ABC: \[ \tan 45^\circ = \frac{\text{Height}}{\text{Base}} = \frac{AB}{BC} \] \[ \Rightarrow 1 = \frac{AB}{BC} \Rightarrow AB = BC \quad \text{(i)} \] From the right-angled triangle ABD: \[ \tan 30^\circ = \frac{AB}{BD} = \frac{1}{\sqrt{3}} \Rightarrow AB = \frac{BD}{\sqrt{3}} \quad \text{(ii)} \] Comparing equations (i) and (ii): \[ BC = \frac{BD}{\sqrt{3}} \Rightarrow \sqrt{3}BC = BD \] \[ \Rightarrow \sqrt{3}BC = BC + CD \] \[ \Rightarrow \sqrt{3}BC - BC = CD \] \[ \Rightarrow BC(\sqrt{3} - 1) = 60 \] \[ \Rightarrow BC = \frac{60}{\sqrt{3} - 1} \] \[ = \frac{60(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{60(\sqrt{3} + 1)}{3 - 1} \] \[ = \frac{60(\sqrt{3} + 1)}{2} = 30(\sqrt{3} + 1) \] Substituting this value into equation (i): \[ AB = BC = 30(\sqrt{3} + 1) \] ∴ The height of the pillar is \(30(\sqrt{3} + 1)\) meters.