Problem 23 - A Complete Resource Book in Mathematics for JEE Main

Question

The angle of elevation of the top of a vertical pole when observed from each vertex of a regular hexagon is \(\frac{\pi}{3}\). If the area of the circle circumscribing the hexagon be \(A\) metre \(^{2}\) then the height of the tower is (A) \(\frac{2 A}{\sqrt{3 \pi}}\) metre (B) \(\frac{A}{\sqrt{3 \pi}}\) metre (C) \(2 \sqrt{\frac{A}{3 \pi}}\) metre (D) \(\sqrt{\frac{A}{3 \pi}}\) metre

Step-by-Step Solution

Answer

Correct answer is not listed. Check for calculation errors or context discrepancy.

Step 1: Understand the Geometry and Observations

The problem involves a vertical pole observed from each vertex of a regular hexagon. The angle of elevation from these vertices to the top of the pole is \( \frac{\pi}{3} \). The area of the circle circumscribing the hexagon is \( A \) square meters.

Step 2: Relate the Side Length and Circumcircle Radius

A regular hexagon can be inscribed in a circle, and its circumcircle has the radius equal to the side length of the hexagon. Since the area \( A \) of the circumscribing circle is \( \pi R^2 \), where \( R \) is the radius, the relationship is \( \pi R^2 = A \). Thus, \( R = \sqrt{\frac{A}{\pi}} \).

Step 3: Determine the Side Length of the Hexagon

For a regular hexagon, the side length \( s \) is equal to the radius of its circumcircle. Therefore, \( s = \sqrt{\frac{A}{\pi}} \).

Step 4: Use Trigonometry to Determine the Height of the Tower

Given that the angle of elevation is \( \frac{\pi}{3} \) from each vertex, we have a right triangle with the pole as the height, and the side of the hexagon as the base. Applying the tangent function, \( \tan\left(\frac{\pi}{3}\right) = \frac{h}{s} \), and since \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \), it follows that \( \sqrt{3} = \frac{h}{\sqrt{\frac{A}{\pi}}} \), or \( h = \sqrt{3} \cdot \sqrt{\frac{A}{\pi}} \).

Step 5: Simplify the Expression for the Height

Simplify \( h = \sqrt{3} \cdot \sqrt{\frac{A}{\pi}} = \sqrt{\frac{3A}{\pi}} \). Thus, the height of the tower is \( \sqrt{\frac{3A}{\pi}} \).

Step 6: Compare with Given Options

Compare the expression \( \sqrt{\frac{3A}{\pi}} \) with the options. Notice none of them matches directly. Check calculations and simplify further if any discrepancies.

Key Concepts

Regular Hexagon

A regular hexagon is a six-sided polygon where all sides and angles are equal. Each internal angle measures 120 degrees, making it a symmetric and uniform shape. One fascinating property of a regular hexagon is its ability to perfectly fit inside a circle, touching the circle at every vertex. This essential geometry is what makes regular hexagons unique when circumscribing circles are involved.
For a regular hexagon, the radius of the circumscribing circle—also called the circumcircle—is equal to the length of one side of the hexagon. This means if you know the radius of the circumcircle, you also know the length of each side of the hexagon. This is a key fact that can help solve many geometry problems involving regular hexagons and circles.
Hexagons are commonly observed in natural structures, such as beehive patterns, due to their efficient use of space.

Angle of Elevation

The angle of elevation is a concept in trigonometry that refers to the angle from the horizontal upward to an object. Imagine standing on flat ground, then looking upwards at a tree or tower; the angle formed between your line of sight and the horizontal is the angle of elevation.
In mathematical problems, the angle of elevation is used to calculate heights and distances. When we use trigonometric functions, particularly the tangent, to determine these measurements, the angle of elevation becomes critically useful. The formula utilizes the relationship between the height of the object and the distance from the observer, encapsulated as:

• Using in our example: - If the angle of elevation is \( \frac{\pi}{3} \) or 60 degrees, we can utilize the tangent function: \( \tan(\frac{\pi}{3}) = \frac{\text{Height}}{\text{Base}} \)

This concept is invaluable when dealing with real-life applications like finding the height of a tower or a mountain by looking from a certain distance.

Circle Area

The area of a circle is a fundamental concept in geometry, determined by the formula \( A = \pi R^2 \), where \( R \) is the radius of the circle. This formula calculates the space enclosed within the circle, defined by its radius.
In solving geometric problems, especially those involving polygons inscribed or circumscribing circles, it's crucial to link their properties with that of circle area. For instance, when a regular hexagon is involved, knowing the area of the circle can help derive other measurements, like the side length, since the side of a regular hexagon equals the circle's radius.
To break it down further:

• The circumference, which is the total distance around the circle, is \( 2\pi R \).
• This basic circle property interplays with the geometry of other shapes, like hexagons, to solve complex problems efficiently.

Understanding this relationship allows students to navigate problems where the features of hexagons and circles overlap.