Question

A surveyor has to measure the horizontal distance from her position to a distant reference point \(C\). Using her position as the center, a 200 m horizontal line segment is drawn with the two endpoints \(A\) and \(B\). Points \(A\), \(B\), and \(C\) are not collinear. Each of the angles \(\angle CAB\) and \(\angle CBA\) are measured as \(87.8^\circ\). The distance (in m) of the reference point \(C\) from her position is nearest to:

• A. 2603
• B. 2606
• C. 2306
• D. 2063

Hint:

For trigonometric problems, use tangent or sine functions effectively for right-angled triangles and small angles.

Answer 1

The Correct Option is A

Step 1: Using the given data, the angles \(\angle CAB = \angle CBA = 87.8^\circ\) and the base \(AB = 200\) m. Step 2: In \(\triangle CAB\), use the tangent function for small angles: \[ \tan \theta = \frac{cx}{AB}, \quad \text{where } cx \text{ is the perpendicular from } C. \] Step 3: Substitute values: \[ \tan(87.8^\circ) = \frac{cx}{100}. \] Step 4: Solve for \(cx\): \[ cx = 100 \times \tan(87.8^\circ) = 2603 \, \text{m}. \] Hence, the correct option is (A).