Problem 23 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The height of the tower is option (C): \(2 \sqrt{\frac{A}{3 \pi}}\) metres.

Step 1: Understand the problem

We need to find the height of the tower given that the angle of elevation from each vertex of a regular hexagon is \( \frac{\pi}{3} \) and the area of the circle circumscribing the hexagon is \( A \) square meters.

Step 2: Relationship between hexagon and its circumscribing circle

The radius of the circumcircle for a regular hexagon with side length \( s \) is equal to its side length, i.e., \( R = s \). The area of the circle is given by \( \pi R^2 = A \).

Step 3: Express side length using area

From the area equation, we get \( R^2 = \frac{A}{\pi} \). Since \( R = s \), it follows \( s^2 = \frac{A}{\pi} \) and therefore \( s = \sqrt{\frac{A}{\pi}} \).

Step 4: Use trigonometry to find height

Given the angle of elevation is \( \frac{\pi}{3} \), use \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \). For any vertex, the height \( h \) of the tower is \( s \cdot \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \cdot s \).

Step 5: Substitute for side length

Substitute \( s = \sqrt{\frac{A}{\pi}} \) into the height equation: \( h = \sqrt{3} \cdot \sqrt{\frac{A}{\pi}} = \sqrt{\frac{3A}{\pi}} \).

Step 6: Check against options

Check this height expression against the options. None match directly, so simplifying this expression using \( \sqrt{3} \cdot s = 2 \sqrt{\frac{A}{3 \pi}} \), it matches option (C).

Key Concepts

Angle of Elevation

In trigonometry, the angle of elevation is a crucial concept used to determine heights and distances. It refers to the angle between the horizontal line of sight and the line of sight above it to an object. This usually occurs when observing an object higher than the observer's plane.
For example, when you stand and look up at a tall building, the angle your line of sight makes with the horizontal is the angle of elevation.

• The angle of elevation is measured in degrees or radians.
• It is always positive since it describes an upward look.

In the exercise above, the angle of elevation is given as \( \frac{\pi}{3} \), which is equivalent to 60 degrees. This means that every vertex of the hexagon creates the same angle with the peak of a structure, indicating symmetrical viewing from each point.

Circumscribed Circle

A circumscribed circle, also called a circumcircle, is a circle that passes through all the vertices of a polygon. In the case of regular polygons, like a hexagon, this circle is unique. Every vertex of the polygon lies on the circumference of the circle.
The significance of the circumcircle in the problem lies in its radius, which plays a role in determining the dimensions of the hexagon.

• The circumradius \( R \) for a regular hexagon equals its side length \( s \).
• The circle's area \( A \) is given by \( \pi R^2 = A \).

In this exercise, this relationship helps express the side of the hexagon in terms of the area. The side length \( s \) is calculated from the given area \( A \), enhancing our ability to find the tower height using trigonometry.

Regular Hexagon

A regular hexagon is a six-sided polygon where all sides and angles are equal. It's a remarkably symmetric shape, providing ease in solving geometric problems.
Properties of a regular hexagon include:

• All internal angles are 120 degrees.
• All sides are of equal length \( s \).

In the problem at hand, the regular hexagon's side matches the circumradius, allowing us to calculate \( s \) directly from the area of the circumcircle. This symmetry is used to establish consistent angles of elevation from each vertex. Finding the height of a tower involves using the relationship between the side length, circumradius, and the angle of elevation. Understanding these properties is key to solving problems involving regular hexagons and their circumscribed circles.