Problem 5 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

An observer finds that the angular elevation of a tower is \(A\). On advancing \(3 \mathrm{~m}\) towards the tower the elevation is \(45^{\circ}\) and on advancing \(2 \mathrm{~m}\) nearer, the elevation is \(90^{\circ}\) \(-A\). The height of the tower is (A) \(2 \mathrm{~m}\) (B) \(4 \mathrm{~m}\) (C) \(6 \mathrm{~m}\) (D) \(8 \mathrm{~m}\)

Step-by-Step Solution

Answer

The height of the tower is 6 m.

Step 1: Set Up the Problem

Write down what is given in the problem: - Initial distance from the tower: unknown, call it 'd'.- Angles of elevation at different positions: A, \(45^{\circ}\), and \(90^{\circ} - A\).- Movements towards the tower: 3 m and then 2 m more.Create equations for the height (h) based on these positions and angles.

Step 2: Use Trigonometry for First Position

From the initial position, the angle of elevation is A.Use the tangent function:\[ \tan(A) = \frac{h}{d} \]Thus, \(h = d \cdot \tan(A)\).

Step 3: Use Trigonometry for Second Position

After moving 3 m farther towards the tower, the angle is \(45^{\circ}\).Use the tangent function:\[ \tan(45^{\circ}) = \frac{h}{d - 3} = 1\]This implies \(h = d - 3\).

Step 4: Use Trigonometry for Third Position

Another advancement of 2 m makes the angle \(90^{\circ} - A\).The tangent of \(90^{\circ} - A\) is the cotangent of A:\[ \tan(90^{\circ} - A) = \frac{h}{d - 5} = \cot(A)\]Thus, \(h = (d - 5) \cdot \cot(A)\).

Step 5: Equate Equations from Different Positions

You now have three equations:1. \(h = d \cdot \tan(A)\)2. \(h = d - 3\)3. \(h = (d-5) \cdot \cot(A)\)Equate 1 and 2:\[ d \cdot \tan(A) = d - 3 \]Solve for d in terms of \(\tan(A)\) and then equate with equation 3.

Step 6: Solve the Combined Equations

Substitute \(d = \frac{3}{1 - \tan(A)}\) back into equation 3:\[ h = \left(\frac{3 - 5(1 - \tan(A))}{1 - \tan(A)}\right) \cdot \cot(A) \]Simplify and solve to find h, knowing common trigonometric identities may help simplify further.

Step 7: Determine Exact Value

Solving the simplified equation results in evaluating for a numerical value of the height that must match one of the options provided. Through algebraic simplification, you will find that h equals 6.

Key Concepts

Tangent Function

The tangent function is one of the basic trigonometric functions in mathematics. It relates the opposite side to the adjacent side of a right triangle. In this problem, the tangent function is key because it helps us find the height of the tower from different positions.
Mathematically, tangent is expressed as:

• \( \tan(A) = \frac{\text{opposite side}}{\text{adjacent side}} \)

Here, the 'opposite side' is the height \( h \) of the tower, and the 'adjacent side' is the distance \( d \) or \( d - 3 \) or \( d - 5 \) from the observer to the tower, depending on her position.
Using the tangent function, we formulated equations for each position using the given angles, crucial in problem-solving when the known variable is the angle, and the unknown is usually the height.

Angle of Elevation

The angle of elevation is the angle between the line of sight from an observer to an object and the horizontal line from the observer. In this exercise, angles of elevation are used to determine the height of a tower.
There are critical angles to consider:

• Initial angle of elevation \( A \)
• Angle of elevation after advancing 3 meters which is \( 45^{\circ} \)
• Angle of elevation after 5 meters in total which is \( 90^{\circ} - A \)

By measuring how the angle of elevation changes as the observer moves closer, one can accurately calculate the height of the structure. Knowing these angles, we used the tangent function to create relationships or equations which are solved to find the desired height.

Problem Solving

Trigonometric functions and angles are crucial in solving problems involving heights and distances. Here is how we applied them to find the tower's height:
1. **Understand the problem:** Before solving, we interpreted the given data — the angles and distances. This helped us create a visual map of the scenario. 2. **Formulate equations:** Using the tangent function, we formulated different expressions for the height of the tower from each position, reflecting how angles vary. 3. **Combine and solve:** By equating these expressions, we reduced the complexity of the problem step by step, eventually solving for the unknowns by using trigonometric identities and algebraic manipulations.
This structured approach is beneficial because it breaks down complex problems into manageable steps. Each mathematical step correlates with a real-world distance or height, improving understanding and accuracy in application.