Problem 26 - A Complete Resource Book in Mathematics for JEE Main

Question

A flagstaff stands verticality on a pillar, the height of the flagstaff being double the height of the pillar. A man on the ground at a distance finds that both the pillar and the flagstaff subtend equal angles at his eyes. The ratio of the height of the pillar and the distance of the man from the pillar is (A) \(\sqrt{3}: 1\) (B) \(1: \sqrt{3}\) (C) \(2: \sqrt{3}\) (D) none of these

Step-by-Step Solution

Answer

The correct ratio is (B) \(1: \sqrt{3}\).

Step 1: Define Variables

Let the height of the pillar be \( h \). Then, the height of the flagstaff is \( 2h \) since it is double the height of the pillar.

Step 2: Calculate Subtended Angle for the Pillar

The pillar and the flagstaff subtend equal angles at the man's eye. Assume the angle subtended by the pillar alone from the base to the top of the pillar is \( \theta \). Using the tangent function, \( \tan(\theta) = \frac{h}{d} \), where \( d \) is the distance from the man to the pillar.

Step 3: Calculate Subtended Angle for the Flagstaff

The total height of the pillar and the flagstaff is \( h + 2h = 3h \). Thus, the angle subtended by the top of the flagstaff from the base is \( \tan(\theta) = \frac{3h}{d} \). Since both sections subtend the same angle, we equate the expressions: \( \frac{h}{d} = \frac{3h}{d'} \), where \( d' \) is the same as \( d \).

Step 4: Solve for Ratio

Given \( \tan(\theta) = \frac{3h}{d} = \frac{h}{d} \), simplify to \( d = \sqrt{3}h \). This implies that the ratio of the height \( h \) of the pillar to the distance \( d \) is \( \frac{h}{d} = \frac{1}{\sqrt{3}} \). This corresponds to \( 1 : \sqrt{3} \).

Step 5: Confirm the Option

The calculated ratio \( 1 : \sqrt{3} \) is option (B). Thus, the correct answer is option (B).

Key Concepts

Subtended Angles

When you look at an object from a distance, the angle between your line of sight and the object is called the subtended angle. This concept comes into play when trying to determine the comparative heights or distances using trigonometry. In this particular problem, both a pillar and a flagstaff standing on top of it subtend the same angle from the perspective of a man standing on the ground.

The subtended angle is crucial here because it helps us equate the two scenarios given in the problem. When the problem mentions equal subtended angles, it implies that the visual size that each segment of the pillar and flagstaff appears to be, from the man's perspective, measures out the same. This equality leads to important trignometrical deductions based on tangential formulas discussed in the solution.

Understanding subtended angles in practical contexts such as this one, where an observer views an object at a distance, can provide intuitive insights into solving real-world measurement challenges.

Tangent Function

In trigonometry, the tangent function is one of the most important tools we have for relating angles to the lengths of sides in right-angled triangles. The tangent of an angle, \(\tan(\theta)\), is defined as the ratio of the opposite side to the adjacent side in a right triangle. This is particularly useful when you are dealing with objects you need to measure indirectly, like in our example with the flagstaff and the pillar.

To find the subtended angle for the pillar, we use \( \tan(\theta) = \frac{h}{d} \), where \( h \) is the height of the pillar and \( d \) is the distance from the observer. For the flagstaff, the total height becomes \( 3h \), leading us to use \( \tan(\theta) = \frac{3h}{d} \). The equality of these two expressions directly informs us about the geometric relationship between these lengths, which in turn allows us to calculate the needed ratio of height to distance.

The tangent function simplifies complex visual data into manageable algebraic expressions, bridging the gap between intuitive observation and precise calculation.

Ratio Calculation

Ratio calculation is a foundational skill in mathematics that helps compare the size, length, or quantity of one thing to another. In the context of this exercise, calculating the ratio between the height of the pillar and the distance of the observer from the pillar allows us to resolve an apparent visual equality into a precise numerical relationship.

After defining that the tangent of the angles are equal, the critical step is to simplify the expression \( \frac{h}{d} \) and \( \frac{3h}{d} \). The simplification and rearrangement of these equations show that \( d = \sqrt{3}h \), which leads us to conclude that the ratio of height \( h \) to distance \( d \) is \( 1 : \sqrt{3} \).

Understanding how to calculate and interpret ratios is crucial not just in trigonometry, but in many real-life applications where comparisons of different magnitudes, heights, or quantities are necessary.