Two poles of equal height stand facing each other along a 40 m wide road. From a point midway on the road, the angles of elevation of the tops of the poles are 60 degrees and 30 degrees, respectively. The heights of the poles are:

1. A

   

2. B

   

3. C

   

4. D

   15m

Appeared in: ESIC Nursing Officer - 2019 (Shift-2)

Explanation

• The problem is modeled using two right-angled triangles with a common height 'h'.
• The base of the triangles are 'x' and '40-x', representing the partitioned width of the road.
• Two trigonometric equations are formed using the tangent function: h = x * tan(60°) and h = (40-x) * tan(30°).
• Solving these simultaneous equations gives the distance x = 10 m.
• Substituting x=10 back into the first equation gives the height h = 10 * tan(60°) = 10√3 m.

Why Other Options Were Wrong

• Option A: This calculation incorrectly assumes the observation point is at the midpoint of the road (20m from each pole). This leads to two different, contradictory values for the height (h = 20√3 and h = 20/√3), violating the condition that the poles are of equal height.
• Option C: Similar to the first option, this calculation is based on the flawed premise that the observation point is at the midpoint (20m from each pole). This is inconsistent with the given angles (60° and 30°) for poles of equal height.
• Option D: This is a numerically incorrect value. The calculated height is 10√3 meters, which is approximately 10 * 1.732 = 17.32 meters, not 15 meters.

Related Visual

• Visual 1: Diagram: Two vertical poles (AB, CD) of equal height 'h' on opposite sides of a horizontal road (BD) of width 40m. A point 'P' is on the road, with distance BP=x and PD=40-x. Lines from P to the tops of the poles (A and C) form angles of elevation 60° and 30° respectively.

Clinical Relevance

• Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Application of trigonometry to solve problems involving heights and distances as background academic context rather than a clinical decision trigger.
• This question tests general aptitude, specifically mathematical reasoning and problem-solving, which are components of many nursing and other professional entrance examinations.
• The ability to perform multi-step calculations and apply logical reasoning is a transferable skill useful in clinical settings for tasks like drug dosage calculations and interpreting data.
• What if? If the problem stated the poles were of different heights, you would need additional information, such as the ratio of their heights or another measurement, to solve for both unknown heights.

How to Approach the Question

• First, carefully read the problem to identify all given information (road width, angles of elevation) and the unknown variable (height of the poles).
• Draw a clear diagram to represent the scenario. Label the poles, the road, the observation point, the height (h), and the distances (x and 40-x).
• Recognize the inconsistency in the prompt ('midway' point with unequal angles) and deduce the correct setup where the point is not at the center.
• Apply the tangent trigonometric ratio (tan θ = opposite/adjacent) to both right-angled triangles formed in your diagram to create two separate equations involving 'h' and 'x'.
• Solve the system of two equations to find the value of 'x' first, and then substitute that value back into either equation to find the final height 'h'.
• Compare your calculated result with the given options to find the correct answer.

Concept Tested & Keywords

• Concept Tested: Application of trigonometry to solve problems involving heights and distances.
• Stem keywords: poles, equal height, road, angles of elevation, 60 degrees, 30 degrees, 40 m
• Lead-in keywords: The heights of the poles are

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