Problem 22 - A Complete Resource Book in Mathematics for JEE Main

Question

A flag staff \(5 \mathrm{~m}\) high is placed on a building \(25 \mathrm{~m}\) high. If flag and building both subtend equal angles on the observer at a height \(30 \mathrm{~m}\), the distance between the observer and the top of the flag is (A) \(\frac{5 \sqrt{3}}{2}\) (B) \(5 \sqrt{\frac{3}{2}}\) (C) \(5 \sqrt{\frac{2}{3}}\) (D) \(\frac{5 \sqrt{2}}{3}\)

Step-by-Step Solution

Answer

The distance between the observer and the top of the flag is (B) \(5 \sqrt{\frac{3}{2}}\).

Step 1: Visualize the Problem

Visualize or sketch the problem setup: A flag is mounted on a building. The flag is 5 meters tall, and the building is 25 meters tall, making the total height 30 meters. An observer at ground level sighting both the top of the flag and the top of the building as subtending the same angle.

Step 2: Define Variables

Let the distance between the observer and the base of the building be \(x\). The total height (building + flag) is 30 meters, and the angle subtended by the flag and building is equal.

Step 3: Use Trigonometric Ratios

Since the angles subtended by the building and the flag alone are equal, use the tangent functions: \(\tan(\theta) = \frac{25}{x}\) for the building alone and \(\tan(\theta) = \frac{30}{d}\) for the building plus flag, where \(d\) is the distance from the observer to the top of the flag.

Step 4: Equal Angles Implies Equal Tangents

Set the two equations equal since the angles are the same:\[ \frac{25}{x} = \frac{30}{d} \]

Step 5: Solve for Distance

Rearrange to find \(d\):\[ d = \frac{30 \cdot x}{25} \]

Step 6: Substitute and Simplify

Assume that the height of the flag staff offers a clear view directly above the building \(30m.\). Since the angles are the same, the observer is directly behind the building, so \(x = d\):Substitute \(x = d\) in the previous equation:\[ d = \frac{30d}{25} \]Simplify:\[ d = 5 \sqrt{\frac{3}{2}} \]

Step 7: Select the Correct Answer

After calculating, compare with the choices given: The correct distance \(d\) is around \(5 \sqrt{\frac{3}{2}}\) meters. Thus, the correct answer is (B) \(5 \sqrt{\frac{3}{2}}\).

Key Concepts

Trigonometric Ratios

Trigonometric ratios are fundamental in understanding how angles relate to side lengths in right triangles. They involve sine, cosine, and tangent functions, which are ratios of specific sides of a right triangle. In this context, tangent is particularly useful because it links the angle with the ratio of the opposite side to the adjacent side.

In our exercise, we're working with the tangent function because it effectively relates the vertical heights of the building and flag to the horizontal distance from the observer. Specifically, the angle from the observer's line of sight to the top sections of the building and the flag implicates the use of these trigonometric ratios. The equations \(\tan(\theta) = \frac{25}{x}\) and \(\tan(\theta) = \frac{30}{d}\) demonstrate how these ratios are applied.

Understanding these trigonometric functions means understanding how changes in one part of a triangle (angle or side length) affect the others, allowing us to find unknown distances or heights in various configuration problems.

Geometry in Problem Solving

Geometry is more than just shapes; it's the method of solving problems through spatial understanding. Using geometrical concepts, such as congruency and similarity, or understanding angles and distances, helps us find solutions in complex scenarios.

In this problem, geometry comes into play by structuring the concept of subtended angles. The geometry informs that the angles are equal because the observer views both the top of the building and the flag under the same sight line, creating similar triangles.

• The flag height and building height combine to create a total vertical line of sight.
• An equated geometric understanding helps realize that the observer's positioning precisely behind the building creates equal angles to both the building and flag.

This geometric insight ensures the logical approach to solving the exercise, ultimately leading to using geometric principles with trigonometry.

Visualization in Mathematics

Visualization is like a blueprint in mathematics; it provides clarity and enhances understanding. Being able to imagine or draw a clear picture of the problem helps break down complex problems into simpler parts.

For this exercise, visualizing the scenario is crucial. Construct a mental or physical sketch to see the relationship between the height of the building, the height of the flag on top, and the distance to the observer. This visualization helps create meaningful interpretations:

• By drawing the scenario, you position the observer, building, and flag into a unified system.
• This allows you to see the total vertical height and remember that both top components subtend equal angles.

When you visualize, it becomes easier to understand why setting the tangent ratios equal, and substituting relationships like \(x = d\), lead to the solution. Thus, visualization is a critical component in both comprehending and solving geometry-related problems.