Question

Two cars are seen from the top of a tower of height 75 m with angles of depression \(30^\circ\) and \(45^\circ\). If the cars are on opposite sides of the tower along the same line, the distance between them is:

• A. \(75(\sqrt{3}+1)\) m
• B. \(75(\sqrt{3}-1)\) m
• C. \(75(\sqrt{3}+1)\) m
• D. \(75(\sqrt{3}-1)\) m

Hint:

In problems involving angles of depression: - Use \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\), where "opposite" is the tower height. - If two objects are on opposite sides, add distances.

Answer 1

The Correct Option is A

Step 1: Use \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \) Let the horizontal distances of cars from the base be \(x\) and \(y\). For angle \(30^\circ\): \[ \tan 30^\circ = \frac{75}{x} \Rightarrow \frac{1}{\sqrt{3}} = \frac{75}{x} \Rightarrow x = 75\sqrt{3} \] For angle \(45^\circ\): \[ \tan 45^\circ = \frac{75}{y} \Rightarrow 1 = \frac{75}{y} \Rightarrow y = 75 \] 
Step 2: Total distance between cars:
\[ \text{Total distance} = x + y = 75\sqrt{3} + 75 = 75(\sqrt{3} + 1) \]