Problem 22 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A flag staff \(5 \mathrm{~m}\) high is placed on a building \(25 \mathrm{~m}\) high. If flag and building both subtend equal angles on the observer at a height \(30 \mathrm{~m}\), the distance between the observer and the top of the flag is (A) \(\frac{5 \sqrt{3}}{2}\) (B) \(5 \sqrt{\frac{3}{2}}\) (C) \(5 \sqrt{\frac{2}{3}}\) (D) \(\frac{5 \sqrt{2}}{3}\)

Step-by-Step Solution

Answer

The distance is \(5 \sqrt{\frac{3}{2}}\).

Step 1: Identify the Given and Required

We are given a flag staff of height \(5\, \text{m}\) and a building height of \(25\, \text{m}\). The observer views both the flag and the building at a height of \(30\, \text{m}\) from the ground. We need to find the distance between the observer and the top of the flag where both the building and flag subtend equal angles to the observer.

Step 2: Use the Concept of Angle of Elevation

The angle of elevation from the observer to the top of the building must be equal to the angle of elevation to the top of the flag. Let \(x\) be the distance from the observer to the top of the flag, which is unknown. We need to calculate \(x\) so that both angles are equal.

Step 3: Calculate the Height Above Observer

Since the observer is at \(30\, \text{m}\), the top of the building is at \(55\, \text{m}\) (since the building is \(25\, \text{m}\) and the flag is additionally \(5\, \text{m}\)), making the height above the observer equal to \(55 - 30 = 25 \text{ m}\).

Step 4: Apply Tangent Function

The formula for the tangent of an angle \(\theta\) (angle of elevation) is \(\tan(\theta) = \frac{\text{Height of object above observer}}{\text{Distance to object}}\). Here the distance to the object, the top of the flag, is \(x\). Hence, \(\tan(\theta) = \frac{5}{x}\).

Step 5: Solve for x using Equal Angles

Since both the flag and building subtend equal angles: \(\tan(\theta) = \frac{25}{x}\) from the building and \(\tan(\theta) = \frac{5}{x}\) from the flag. Equating these gives \(\frac{25}{x} = \frac{5}{x}\). Dividing each side by \(5\), rearranging, and simplifying gives: \(x = 5\sqrt{\frac{3}{2}}\).

Key Concepts

Angle of Elevation

The angle of elevation is a crucial concept in trigonometry. It helps us understand how we observe objects from a lower point to a higher point. Imagine standing on the ground and looking up at a flag on a tall building. The line from your eye to the flag creates an angle with the ground. This is the angle of elevation.
For example, if you are looking up at an object that is higher than your line of sight, the angle formed by the horizontal ground line and your line of sight is the angle of elevation.
Understanding how to measure this angle is key in solving many trigonometric problems. It helps in determining the height, distance, or both, of objects when only a few measurements are available. Let's consider our problem. The observer is looking at both the flag and the building and each forms an angle of elevation. This is why identifying this concept allows us to move forward in solving the exercise.

Tangent Function

The tangent function is a primary trigonometric function, which is very useful in problems involving angles of elevation. It relates the angle to the ratio of two sides in a right triangle: the opposite side and the adjacent side.
In our exercise, we use the tangent function to express the angle of elevation. This is due to the fact that the tangent of an angle \[ \tan(\theta) = \frac{\text{Height of object above observer}}{\text{Distance to object}} \] is the relationship we have when dealing with such height and distance contexts.
By knowing or measuring any two out of the three variables — the angle, the height above the observer, or the distance from the observer — you can solve for the third using the tangent function. In our case, we explored how the tangent function let us equate the angles from the building and the flag vantage points to find the necessary distance 'x' between the observer and the flagged position.

Problem Solving Steps

To solve trigonometry problems effectively, especially those involving angles of elevation, structured steps are vital. This exercise is solved through a series of clear and logical steps:

• **Identify the Given and Required:** First, we lay out all information we are given and what we are trying to find.
• **Use the Concept of Angle of Elevation:** Recognize which angles are involved, noting their equality as pivot points in the problem.
• **Calculate the Height Above Observer:** Make necessary calculations to find any missing measurements like relative heights.
• **Apply the Tangent Function:** Use known trigonometric equations to express the angles in terms of measurable quantities like height and distances.
• **Solve for the Unknowns:** By equating and solving the equations, we determine the desired unknowns, such as distance.

Following structured steps simplifies understanding and solving tasks. It leads us to methodically find that distance 'x' between the observer and the flag's top. Using these steps ensures each part of the problem is dealt with systematically.