Problem 45 - A Complete Resource Book in Mathematics for JEE Main 2017

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) 7a \(\tan \alpha\)

Step-by-Step Solution

Answer

The height of the tower is given by option (A): \(3a \tan \alpha\) or \(5a \tan \beta\).

Step 1: Analyze the First Observation

Initially, the person stands at the foot of the tower and walks a distance of \(3a\) away from the tower. At this point, the angle of elevation is \(\alpha\). We express the height of the tower \(h\) using basic trigonometry as \( h = 3a \tan \alpha \).

Step 2: Analyze the Second Observation

Next, the person walks an additional \(4a\) distance perpendicular to his initial path. At this new point, the angle of elevation changes to \(\beta\). Using trigonometry again, the horizontal distance from the tower in relation to this angle is calculated using the Pythagorean theorem: \( (3a)^2 + (4a)^2 = 25a^2 \). Therefore, the total distance is \(5a\). Thus, the height of the tower can also be expressed as \( h = 5a \tan \beta \).

Step 3: Compare the Expressions for Height

We now have two expressions for the height of the tower: \( h = 3a \tan \alpha \) (from the first observation) and \( h = 5a \tan \beta \) (from the second observation). The tower height must satisfy both equations.

Step 4: Evaluate the Options

Looking at the given options, option A states the height as \(3a \tan \alpha\) or \(5a \tan \beta\). This matches the derived expressions in step 3 for the tower's height. The other options are incorrect based on the derived expressions.

Key Concepts

Angle of Elevation

When you look at something higher than your eye level, like the top of a tower, you're experiencing what we call the "angle of elevation." This angle is the spot between your line of sight as you look at the object and the horizontal line from where you are standing. It's like you're tilting your head up.
For instance, in the problem, a person looked at the top of a tower. The angle their line of sight makes with the horizontal is called the "angle of elevation." An important point to remember is that the angle of elevation changes as you move closer or further away from the tower.
The angle of elevation helps determine distances and heights when combined with trigonometric concepts. By knowing the angle of elevation and the distance from the tower, you can use the tangent function to find the height of the tower.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept that helps find the lengths of sides in a right triangle. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of squares of the other two sides. The formula is \[ c^2 = a^2 + b^2 \].
In the exercise, the theorem is used when the person walks at a distance perpendicular to the initial path, creating a right triangle. The two perpendicular distances, \(3a\) and \(4a\), form the legs of a triangle.
Using this setup, the hypotenuse is calculated: \( (3a)^2 + (4a)^2 = 25a^2 \), which simplifies to \(5a\) as the hypotenuse. This step is vital for finding the distance from which the angle \(\beta\) is observed.

Trigonometric Ratios

Trigonometric ratios are ratios derived from the angles and sides of a right triangle, essential in solving problems involving heights and distances. The key ratios are sine (sin), cosine (cos), and tangent (tan). Each ratio involves different sides of a right triangle.
The tangent of an angle, \( \theta \), is the ratio of the opposite side to the adjacent side, written as \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).
In this scenario, tangent is used to find the tower's height. Initially, when the person looks at the tower from \(3a\) distance, the tangent ratio helps express the height as \(h = 3a \tan \alpha\). Similarly, after the person moves further, the height can also be expressed as \(h = 5a \tan \beta\). The consistency in these measurements confirms the tower's height under different viewpoints.