Problem 46 - A Complete Resource Book in Mathematics for JEE Main

Question

A stationary balloon is observed from three points \(\mathrm{A}\), \(\mathrm{B}\) and \(\mathrm{C}\) on the plane ground and is found that its angle of elevation from each point is \(\alpha\). If \(\angle \mathrm{ABC}=\beta\) and \(\mathrm{AC}=\mathrm{b}\), the height of the balloon is (A) \(\frac{b}{2} \tan \alpha \operatorname{cosec} \beta\) (B) \(\frac{b}{2} \tan \alpha \sin \beta\) (C) \(\frac{b}{2} \cot \alpha \operatorname{cosec} \beta\) (D) \(\frac{b}{2} \cot \alpha \sin \beta\)

Step-by-Step Solution

Answer

Option (A) \( \frac{b}{2} \tan \alpha \operatorname{cosec} \beta \).

Step 1: Understanding the Question

The problem provides that a balloon is observed from three points A, B, and C with the same angle of elevation \( \alpha \). Additionally, \( \angle ABC = \beta \) and the distance \( AC = b \). We are to find the height of the balloon related to these angles and this distance.

Step 2: Analyzing Triangle Geometry

For points A, B, and C on the ground and at a height where the balloon is, triangle ABC projects two right triangles, namely \( \triangle AHB \) and \( \triangle CHB \), where H is the point directly below the balloon. Both triangles have the height (h) we need to find as a vertical side.

Step 3: Applying Trigonometric Identities

Since the angle of elevation from A and C is \( \alpha \), employ \( h = AB \cdot \tan \alpha \) and \( h = BC \cdot \tan \alpha \). Notice also that by sine rule \( \frac{AB}{\sin \beta} = \frac{AC}{\sin(90^\circ - \beta)} = \frac{b}{\cos \beta} \).

Step 4: Substituting for AB and BC

Substitute \( AB = \frac{b}{\cos \beta} \sin \beta \) into \( h = AB \cdot \tan \alpha \) gives \( h = \frac{b \cdot \tan \alpha \cdot \sin \beta}{2} \cdot \csc \beta \). This formula simplifies to \( h = \frac{b \cdot \tan \alpha}{2} \cdot \csc \beta \).

Step 5: Selection of Correct Option

After simplifying the equation, observe that this matches option (A). Hence, the height of the balloon is \( \frac{b}{2} \tan \alpha \operatorname{cosec} \beta \).

Key Concepts

Angle of Elevation

Imagine you're on the ground looking up at a balloon. The angle your line of sight makes with the ground is called the angle of elevation. This concept is crucial in trigonometry to determine heights of objects like the balloon.
When you move further back or closer to the balloon, the angle of elevation changes. A larger angle of elevation indicates that you are closer to the base of the object which is above you.
This angle is very straightforward:

• If the balloon is directly overhead, the angle is 90 degrees.
• If you are very far away, approaching the horizon, the angle approaches 0 degrees.

Understanding the angle of elevation helps in applying right triangle trigonometry effectively. You can then use these angles with functions like sine, cosine, and tangent to solve for unknown lengths, making this concept invaluable in practical applications and exercises.

Sine Rule

The Sine Rule is useful in any triangle, not just right triangles. It helps to find unknown sides or angles given some initial information.
This rule states: For any triangle with angles and sides, the ratio of the length of a side to the sine of its opposite angle is constant.
Here's how it looks mathematically:

• In triangle ABC, \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

In the balloon example, the sine rule is applied to understand the relationship between side AC and angle ß. This provides a useful way to find unknown lengths, like using \[AB = \frac{b}{\cos \beta} \sin \beta\]to solve for parts of the triangle not immediately clear from just angles.
This showcases the power of trigonometric identities and allows us to analyze non-right triangles effectively.

Right Triangles

Right triangles form the backbone of trigonometric studies. They have one 90-degree angle, giving them unique properties that allow us to use basic trigonometric ratios confidently.
In a right triangle, the side opposite to the right angle is the hypotenuse. The other two sides are the adjacent and opposite sides relative to the angle you're evaluating.
Trigonometry heavily relies on these formulas:

• \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)
• \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)
• \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)

For the balloon problem, the right triangle concept helps calculate height using tangent \[h = AB \cdot \tan \alpha\].Understanding these basics allows problem-solving by relating angles and distances, showing the strength of trigonometry in practical use and applications.