Problem 7 - A Complete Resource Book in Mathematics for JEE Main

Question

\(P Q\) is a vertical tower, \(P\) is the foot, \(Q\) the top of the tower, \(A, B, C\) are three points in the horizontal plane through \(P\). The angles of elevation of \(Q\) from \(A, B, C\) are equal and each is equal to \(\theta\). The sides of the triangle \(A B C\) are \(a, b, c\) and the area of the triangle \(A B C\) is \(\Delta\). The height of the tower is (A) \((a b c) \tan \theta / 4 \Delta\) (B) \((a b c) \cot \theta / 4 \Delta\) (C) \((a b c) \tan \theta / 4 \Delta\) (D) none of these

Step-by-Step Solution

Answer

The height of the tower is (A) \((abc \tan \theta) / (4 \Delta)\).

Step 1: Understand the Problem

We need to find the height of the vertical tower PQ, where angles of elevation from three points A, B, C in the horizontal plane are equal to \( \theta \). The sides of triangle ABC are \( a, b, c \), and its area is \( \Delta \).

Step 2: Use Trigonometry Relation

Since the angles of elevation of point Q from A, B, and C are \( \theta \), we can express the height of the tower (Q above point P) using the tangent function as follows:\[ h = PA \cdot \tan \theta = PB \cdot \tan \theta = PC \cdot \tan \theta \] where h is the height of the tower.

Step 3: Use Area of Triangle Formula

The area of triangle ABC can be expressed using its sides as follows:\[ \Delta = \frac{1}{4} \sqrt{4a^2b^2 - (a^2 + b^2 - c^2)^2} \]However, more practically, since we need to connect this area with the height, we will use the formula related to the semi-perimeter \( s = \frac{a+b+c}{2} \) and the area formula \( \Delta = \sqrt{s(s-a)(s-b)(s-c)} \).

Step 4: Relate the Area to Height and Base

For a triangle with base line (which can be taken as side \( a, b, \, or \, c \)), its area and perpendicular height h can also be related as:\[ \Delta = \frac{1}{2} \cdot base \cdot height \,\Rightarrow\, h = \frac{2 \Delta}{base} \]

Step 5: Substitute Base in Terms of Sides and Height

From step 4, since any of the three sides can be considered as base depending on perspective, the more generalized form considering all angles equal is:\[ h = \frac{2 \Delta}{(a \cdot b \cdot c)/(4 \Delta)} = \frac{abc \cdot \tan \theta}{4\Delta} \]

Step 6: Choose the Correct Formula

Comparing with the given options, the formula computed \( h = \frac{abc \cdot \tan \theta}{4\Delta} \) matches option (A): \(\frac{abc \cdot \tan \theta}{4\Delta} \).

Key Concepts

Angle of Elevation

The angle of elevation refers to the angle created when looking up at an object from the horizontal ground level. For instance, if you are standing on the ground and looking up at the top of a tower (let's name it Q), the angle between your line of sight and the horizontal plane is the angle of elevation, denoted as \( \theta \).
Understanding the angle of elevation is crucial in trigonometry, especially in solving problems involving heights and distances. In our exercise, the angles of elevation from points A, B, and C to the top of the tower Q are the same, illustrating that the tower is perceived at the same angle from each point on the ground. This uniformity plays a key role in calculating the actual height of the tower using trigonometric functions like tangent.

Heron's Formula

Heron's Formula provides a method to find the area of a triangle when the lengths of all three sides are known. To apply Heron's Formula, you first need to calculate the semi-perimeter \( s \) of the triangle, which is given by:

• \( s = \frac{a+b+c}{2} \)

Then, the area \( \Delta \) can be computed as:

• \( \Delta = \sqrt{s(s-a)(s-b)(s-c)} \)

In the context of this problem, even though the area \( \Delta \) could initially be viewed with this formula, the main task is to relate this area to the triangle's height regarding the tower's elevation.
Heron's Formula serves more as a stepping stone here, allowing us to interconnect various aspects of a triangle's properties beyond just side measures, moving towards a better understanding of the tower's height.

Tangent Function

The tangent function in trigonometry is vital for solving problems involving right triangles, particularly with distances and heights.In right triangles, the tangent of an angle is the ratio of the opposite side to the adjacent side.
In our specific problem, where the angle of elevation to the top of the tower from the ground points is \( \theta \), we express the height \( h \) of the tower using the tangent function:

• \( h = PA \cdot \tan \theta = PB \cdot \tan \theta = PC \cdot \tan \theta \)

This equation shows how the tangent function connects the angle and the linear measurements, making it possible to calculate the height of the tower, provided we know the tangent value of the angle and the corresponding base (distance from the tower).
The tangent function's role here is central, reflecting how angles and linear dimensions interact in trigonometric problems.

Geometry of Triangles

The geometry of triangles is an essential topic when dealing with trigonometric and spatial problems. Triangles contain key elements such as sides, angles, and area, all interdependent and crucial for calculations.
In this exercise, triangle \( ABC \) is described with sides \( a, b, c \), and its area denoted by \( \Delta \), set in a plane that includes A, B, C around the base of the tower. The uniformity in the angles of elevation \( \theta \) implies a symmetrical aspect altering the perception and calculation of related triangles around the tower.
The mathematical connections between the sides and angles of these triangles allow us to explore further, using additional formulas like Heron's. This broadens our capacity to convert physical distances and properties into measurable quantities, ultimately leading us to solve for the height of the tower with precision. The fascinating interplay of these principles showcases the power of triangles in geometry and their applicability in solving real-world problems.