Question

X and Y are two points that are 135 m apart on the ground on either side of a pole and in the same line. The angles of elevation of a bird sitting on the top of the pole from X and Y are \( 30^\circ \) and \( 60^\circ \) respectively. The distance of Y from the foot of the pole (in m) is:

• A. 50.63
• B. 33.75
• C. 67.5
• D. 101.25

Hint:

Use basic trigonometry (tan θ = opposite/adjacent) and set up equations. Relating two expressions for height often helps eliminate the variable.

Answer 1

The Correct Option is C

Let height of pole be \( h \), distance from X be \( x \), and distance from Y be \( y \). From X: \[ \tan 30^\circ = \frac{h}{x} \Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{x} \Rightarrow h = \frac{x}{\sqrt{3}} \] From Y: \[ \tan 60^\circ = \frac{h}{y} \Rightarrow \sqrt{3} = \frac{h}{y} \Rightarrow h = y\sqrt{3} \] Equating: \[ \frac{x}{\sqrt{3}} = y\sqrt{3} \Rightarrow x = 3y \] Now, \[ x + y = 135 \Rightarrow 3y + y = 135 \Rightarrow 4y = 135 \Rightarrow y = 33.75
\text{So distance from Y = } y = 67.5 \]