Question

From a helicopter, a person observes an object on the ground at an angle of depression of \(30^\circ\). If the helicopter is flying at a height of 500 m from the ground, then the distance between the person and the object is:

• A. 500 m
• B. 1000 m
• C. \(500\sqrt{2}\) m
• D. \(500/\sqrt{3}\) m

Hint:

When working with angles of depression, use the tangent function \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\) to calculate the distance between the observer and the object.

Answer 1

The Correct Option is C

The angle of depression and the angle of elevation are equal. Thus, the angle of depression is \(30^\circ\). Using trigonometry, we can relate the height of the helicopter, the distance to the object on the ground, and the distance between the person and the object using the tangent function: \[ \tan(30^\circ) = \frac{\text{height}}{\text{distance}} \] Let the distance be \(d\). Therefore: \[ \tan 30^\circ = \frac{500}{d} \] Since \(\tan 30^\circ = \frac{1}{\sqrt{3}}\), we have: \[ \frac{1}{\sqrt{3}} = \frac{500}{d} \quad \Rightarrow \quad d = 500\sqrt{3} \, \text{m} \] Thus, the correct answer is option (3).