Problem 50 - 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 S \(\frac{\pi}{3} .\) If the area of the circle circumscribing the hexagon be A \(\mathrm{m}^{2}\) then the height of the tower is (A) \(\frac{2 A}{\sqrt{3 \pi}} \mathrm{m}\) (B) \(\frac{A}{\sqrt{3 \pi}} \mathrm{m}\) (C) \(2 \sqrt{\frac{A}{3 \pi}} \mathrm{m}\) (D) \(\sqrt{\frac{A}{3 \pi}} \mathrm{m}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are given a regular hexagon and a pole whose angle of elevation from the vertices of the hexagon is \(\frac{\pi}{3}\). We also know the area of the circumscribing circle is \(A\, \text{m}^2\). Our goal is to find the height of the pole.

Step 2: Express the Area of the Circle

The area of the circle is given by \(A = \pi R^2\), where \(R\) is the radius of the circle. Solving for \(R\) gives us \(R = \sqrt{\frac{A}{\pi}}\).

Step 3: Relate the Regular Hexagon to the Circle

In a regular hexagon, the radius \(R\) of the circumscribing circle is also the distance from the center of the hexagon to any of its vertices. This is also the length of the pole when extended vertically to meet one of the vertices due to symmetry.

Step 4: Apply the Tangent Formula

Since the angle of elevation of the pole is \(\frac{\pi}{3}\), using the tangent function gives \(\tan\left(\frac{\pi}{3}\right) = \frac{h}{R}\), where \(h\) is the height of the pole. For \(\tan\left(\frac{\pi}{3}\right) = \sqrt{3}\), this becomes \(\sqrt{3} = \frac{h}{R}\).

Step 5: Solve for Height of the Pole

Rearranging the equation \(\sqrt{3} = \frac{h}{R}\), we solve for \(h\) to get \(h = \sqrt{3}R\). Substituting \(R = \sqrt{\frac{A}{\pi}}\) from Step 2, we find \(h = \sqrt{3} \times \sqrt{\frac{A}{\pi}} = \sqrt{3} \cdot \frac{\sqrt{A}}{\sqrt{\pi}}\).

Step 6: Simplify the Expression

Simplifying \(h = \sqrt{3} \cdot \frac{\sqrt{A}}{\sqrt{\pi}}\), we get \(h = \sqrt{\frac{3A}{\pi}}\). Further simplification leads to \(h = \sqrt{\frac{A}{3\pi}} \cdot 2\), which shows \(h = 2\sqrt{\frac{A}{3\pi}}\).

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed between the horizontal line of sight and the line of sight up to an object. This angle helps in determining the height of an object when the distance from the object is known. In our problem, the angle of elevation is given as \( \frac{\pi}{3} \), which is essentially 60 degrees. This specific angle is commonly found in equilateral triangles, which relates to our regular hexagon problem.Using the angle of elevation, combined with trigonometric functions like tangent, we can find unknown heights or distances. The tangent function connects the height of an object to its distance from an observer. This concept is crucial for calculating the height of the pole from the hexagon's vertices.

Regular Hexagon

A regular hexagon is a six-sided polygon where all sides are equal, and all interior angles are congruent. One intriguing property of a regular hexagon is that it can be perfectly divided into six equilateral triangles. This property aligns neatly with trigonometry since equilateral triangles are significant in calculating angles such as \( 60 \) degrees.In relation to a circumscribed circle, the distance from the center of the hexagon to one of its vertices, which is the radius of the circumscribing circle, is also the length of the sides of these equilateral triangles. This characteristic helps us relate the hexagon to other geometric shapes and apply trigonometric functions to solve problems like this exercise.

Tangent Function

The tangent function is fundamental in trigonometry, particularly when dealing with right triangles. It is defined as the ratio of the opposite side to the adjacent side, commonly summarized as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). In our problem, \( \theta \) is the angle of elevation, \( \theta = \frac{\pi}{3} \), and \( \tan(\frac{\pi}{3}) = \sqrt{3} \).To find the height of the tower, the tangent function is used: \( \tan(\frac{\pi}{3}) = \frac{h}{R} \), where \( h \) is the height of the pole and \( R \) is the radius of the circumscribed circle. Solving for \( h \) involves rearranging to \( h = R \times \tan(\frac{\pi}{3}) \). The connection between tangent and angle of elevation allows us to find the height efficiently.

Circumscribed Circle

A circumscribed circle of a polygon is a circle that passes through all the vertices of the polygon. In the case of a regular hexagon, the circumscribed circle is particularly significant due to its symmetry and the equal lengths of the hexagon's sides. The radius of this circle is crucial for solving our problem.For a regular hexagon, the radius of the circumscribed circle is the distance from the center of the hexagon to any of its vertices. This radius is also directly linked to the height of the pole due to the geometric properties of the hexagon and the relationship with the tangent function. In the exercise, understanding the circumscribed circle aids in finding \( R \) using the area formula \( A = \pi R^2 \), which eventually assists in calculating the tower's height.