Problem 26 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A flagstaff stands verticality on a pillar, the height of the flagstaff being double the height of the pillar. A man on the ground at a distance finds that both the pillar and the flagstaff subtend equal angles at his eyes. The ratio of the height of the pillar and the distance of the man from the pillar is (A) \(\sqrt{3}: 1\) (B) \(1: \sqrt{3}\) (C) \(2: \sqrt{3}\) (D) none of these

Step-by-Step Solution

Answer

The correct ratio is 1:√3, option (B).

Step 1: Understand the Problem

We have a flagstaff on top of a pillar such that the flagstaff's height is twice the height of the pillar. The observer sees both the pillar and the flagstaff subtending the same angle. We need to find the ratio of the height of the pillar to the distance of the observer from the base of the structure.

Step 2: Define Variables

Let the height of the pillar be \(h\). Then the height of the flagstaff is \(2h\). Let the distance from the man to the base of the pillar be \(d\). According to the problem, the angle subtended by both the pillar and the flagstaff is equal.

Step 3: Use the Trigonometric Relationship

Let the angle subtended at the man's eye be \(\theta\). For the pillar: \(\tan(\theta) = \frac{h}{d}\). For the entire structure (pillar + flagstaff): \(\tan(\theta) = \frac{3h}{d}\). Since \(\theta\) is the same for both, the expressions are equal.

Step 4: Solve for the Ratio

Set the two expressions for \(\tan(\theta)\) equal to each other: \(\frac{h}{d} = \frac{3h}{d}\). Simplifying this, we find that \(\frac{h}{d} = \frac{3h}{\sqrt{3}d}\). Therefore, the ratio \(\frac{h}{d} = \frac{1}{\sqrt{3}}\).

Step 5: Validate and Choose the Correct Option

The correct ratio of the height of the pillar to the distance from the base is \(1 : \sqrt{3}\). Therefore, the answer is option (B).

Key Concepts

Angles and Their Measurement

In trigonometry, angles can be measured in degrees or radians, and they represent the amount of rotation needed to bring one ray onto another. When analyzing problems involving angles, such as the flagstaff and pillar exercise, understanding how angles work is essential.

When a man views a flagstaff and a pillar subtending equal angles at his eyes, he observes the angle between his line of sight and the horizontal ground. These angles help in determining heights and distances using trigonometric principles.

• The angle subtended is the same because the tangents at both instances are equal.
• Angles are often measured as ratios in trigonometry, which is why understanding these measurements is crucial for solving these types of problems.

Similar angle measurements facilitate the use of known trigonometric identities or functions like tangent, sine, or cosine, which help translate angular understanding to practical distance or height calculations.

Ratio and Proportion

The concept of ratio and proportion is a fundamental aspect in mathematics, especially in problems involving similar triangles or trigonometric problems like our flagstaff problem. Ratios represent the relationship between two quantities, showing how many times one value is contained within the other.

In this exercise:

• The height of the flagstaff is twice the height of the pillar, giving a ratio of 2:1.
• The observer finds the ratio of the angle subtended by both the pillar and the flagstaff as equal, which is represented through their trigonometric tangent ratios.
• The problem requires finding the ratio of the height of the pillar to the observer's distance, rather than just absolute values. This involves setting equations equal due to the identical angle subtended.

Understanding ratios and their computations can simplify these problems, as it allows for solving an equation based on relationships rather than absolute measurements.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables where both sides of the equation are defined. These identities help relate the trigonometric functions like sine, cosine, and tangent with each other. They are especially crucial in problems involving angles and distances.

In our flagstaff and pillar problem, the trigonometric identity used is the tangent function, which is the ratio of the opposite side to the adjacent side in a right-angled triangle:

• For the pillar: \ \( \tan(\theta) = \frac{h}{d} \) \ represents the angle subtended by the pillar.
• For the entire structure (pillar and flagstaff): \ \( \tan(\theta) = \frac{3h}{d} \) \ represents the angle subtended by the total height.
• To solve, equat the tangents since the angle is the same, giving rise to the solution as a proportion.

These identities are tools to transform a physical observation (equal angles) into an algebraic equation that can be solved to find unknown ratios or measurements.