Problem 45 - A Complete Resource Book in Mathematics for JEE Main

Question

A person standing at the foot of a tower walks a distance \(3 \mathrm{a}\) away from the tower and observes that the angle of elevation of the top of the tower is \(\alpha\). He then walks a distance \(4 \mathrm{a}\) perpendicular to the previous direction and observes the angle of elevation to be \(\beta\). Then height of the tower is (A) \(3 \mathrm{a} \tan \alpha\) or \(5 \mathrm{a} \tan \beta\) (B) \(5 \mathrm{a} \tan \alpha\) or \(3 \mathrm{a} \tan \beta\) (C) \(4 \mathrm{a} \tan \beta^{\prime}\) (D) \(7 \mathrm{a} \tan \alpha\)

Step-by-Step Solution

Answer

Answer is (A): \(3a \tan \alpha\) or \(5a \tan \beta\).

Step 1: Understand the Scenario

Visualize the scenario: There's a tower at point A. The observer starts at the base (point D), walks 3a units in one direction (point B), then makes a 90° turn and walks 4a units perpendicular to the first direction (point C). The angles \( \alpha \) and \( \beta \) are observed from B and C respectively.

Step 2: Relate Distances Using Trigonometry

Model the problem using trigonometry. At point B (3a distance), using \( \tan \alpha \), we have: \[ \tan \alpha = \frac{\text{Height of tower (h)}}{3a} \]At point C (4a more in perpendicular direction), using \( \tan \beta \), we have: \[ \tan \beta = \frac{h}{\sqrt{(3a)^2 + (4a)^2}} \]

Step 3: Calculate the Compound Distance (Using Pythagorean Theorem)

Calculate the straight-line distance from C to the tower using the Pythagorean theorem. \[ \text{Distance} = \sqrt{(3a)^2 + (4a)^2} = \sqrt{9a^2 + 16a^2} = \sqrt{25a^2} = 5a \]

Step 4: Equate Expressions for the Tower Height

Equate the expressions for height h from both points B and C:For point B: \[ h = 3a \tan \alpha \]For point C:\[ h = 5a \tan \beta \]

Step 5: Validate the Options

From Step 4, it's evident that both expressions for height h lead us to option A, which states the height can be either \( 3a \tan \alpha \) or \( 5a \tan \beta \). This matches the equations derived.

Key Concepts

Angle of Elevation

Imagine you are standing on flat ground, looking up at the top of a tower. The angle that your line of sight makes with the horizontal ground is called the **angle of elevation**. It's like tilting your head up to see a tall building.
If you're closer to the tower, that angle is larger; if you're farther away, the angle is smaller.
This angle can be measured if you know the distance from you to the base of the tower and the tower's height using **trigonometry**. By using the tangent function, which is one of the trigonometric ratios, you can find this angle.

• The formula is: \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)
• Where: **opposite** is the height of the tower, and **adjacent** is the distance from the base of the tower.

This formula is very useful for real-life problems where you need to decide the height of tall structures.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical principle that helps to find the distance between two points in a right-angled triangle. In our scenario with the tower, the observer walks at right angles from one point to another. This situation forms a right triangle.
When you have a right-angled triangle with sides a, b, and hypotenuse c, the Pythagorean Theorem states:
\[ c^2 = a^2 + b^2 \]
In our tower problem, after the person walks 3a units and then makes a 90-degree turn to walk 4a more units, the distance from the new point to the tower can be calculated using this theorem, resulting in a hypotenuse of 5a.

• This theorem helps verify and calculate the longest side of the right triangle (hypotenuse), crucial for understanding trigonometric applications.
• It simplifies determining distances when directions or path changes occur."

This theorem is a building block for many geometry problems and analyzing spatial movement.

Trigonometric Ratios

Trigonometric ratios are essentially relationships between the angles and sides of right-angled triangles. These ratios provide powerful tools when dealing with triangles, especially in problems involving heights and distances.
There are three primary trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). Each of these ratios compares two sides of a right triangle:

• **Sine** relates the opposite side to the hypotenuse: \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
• **Cosine** relates the adjacent side to the hypotenuse: \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• **Tangent** relates the opposite side to the adjacent side: \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).

In our problem, the tangent ratio is crucial. By observing the angles \( \alpha \) and \( \beta \) from different positions, the tangent function helps to determine the height of the tower depending on the observer's distance.
By understanding and applying these trigonometric ratios, you can solve various problems in trigonometry with confidence and clarity.