Problem 59 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A person standing on the ground observes the angle of elevation of the top of a tower to be \(30^{\circ}\). On walking a distance a in a certain direction he finds the elevation of the top to be the same as before. He then walks a distance \(5 \mathrm{a} / 3\) at right angles to his former direction, and finds that the elevation of the top has doubled. The height of the tower is (A) \(\sqrt{5 / 6} a\) (B) \(\sqrt{85 / 48} a\) (C) \(\sqrt{6 / 5} a\) (D) \(\sqrt{48 / 85} a\)

Step-by-Step Solution

Answer

The correct height of the tower is option (B) \(\sqrt{85/48} a\).

Step 1: Understand the Problem

We are given that a person observes the top of a tower at an angle of elevation of \(30^\circ\). After walking a distance \(a\) in the initial direction, the angle remains \(30^\circ\). After turning 90 degrees and walking \(\frac{5a}{3}\), the angle doubles to \(60^\circ\). We need to find the height of the tower.

Step 2: Use the Initial Condition with Trigonometry

The initial observation with a \(30^\circ\) angle suggests a right triangle where \(\tan(30^\circ) = \frac{h}{d}\), with \(h\) being the height of the tower and \(d\) the distance from the tower. Thus, \(h = d \times \tan(30^\circ) = \frac{d}{\sqrt{3}}\) since \(\tan(30^\circ)=\frac{1}{\sqrt{3}}\).

Step 3: Analyze the Impact of Walking

After walking distance \(a\), the angle remains \(30^\circ\). This means the height \(h\) in relation to any distance change remains consistent. Thus, the only change comes from altering the geometry of the situation.

Step 4: Adjust for New Position with Right Angle Shift

He now walks \(\frac{5a}{3}\) at a right angle. Now, we have \(h=\frac{d_2}{3}\), where \(d_2\) is the new horizontal distance and \(\tan(60^\circ) = \sqrt{3}\). The new position forms a triangle with sides \(d+a\) and the perpendicular \(\frac{5a}{3}\) and hypotenuse given by Pythagoras as \(\sqrt{(d+a)^2 + (\frac{5a}{3})^2}\).

Step 5: Equate and Solve for h

From the triangle \(h = \frac{\sqrt{3} \times d_2}{3}\). Solving gives \(d_2 = \sqrt{(d+a)^2 + \left(\frac{5a}{3}\right)^2}\). Substitute for \(d\) and solve accordingly to obtain height \(h\) in terms of \(a\). After substitution and simplification, we solve for \(h\) which corresponds to one of the options provided.

Key Concepts

Angle of Elevation

The concept of the angle of elevation is essential in understanding many real-world problems involving heights and distances. It refers to the angle between the horizontal line of sight and the line of sight up to an object. This angle is commonly observed when you look up at something, like the top of a tower from the ground.

In trigonometry problems, determining the angle of elevation helps in forming a right triangle where one can use trigonometric functions to calculate various unknowns like the height of an object or the distance to the object. It's important to visualize this concept as part of the larger triangle formed by you, the object's top, and the ground when you're solving these types of problems.

• Angle of elevation increases as you get closer to the object.
• For any given height, the angle changes based on your position relative to the base.

Understanding this angle helps you set up and solve equations that are essential for determining the dimensions in trigonometry problems.

Right Triangle

A right triangle is a triangle where one angle is exactly 90 degrees. This type of triangle is crucial for solving many problems in trigonometry because it allows the use of specific trigonometric functions that relate the sides and angles of the triangle.

In the problem given, the scenario involves creating right triangles with the tower height and the observer's distance. The person on the ground and the top of the tower form the hypotenuse, while the height of the tower is the opposite side relative to the angle of elevation given.

• The Pythagorean theorem is key: it relates the lengths of the sides of a right triangle (a² + b² = c²).
• Using right triangles allows for applications of trigonometric functions like sine, cosine, and tangent.

Without forming a right triangle in these problems, it becomes difficult to apply standard trigonometric calculations to find unknown distances or heights.

Tangent Function

The tangent function plays a pivotal role when dealing with angles of elevation in right triangles. The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side.

In the trigonometry problem of a tower's height, the tangent function helps to relate the angle of elevation to the known distances. So, if you know the distance from the base and the angle, you can determine the height of the tower as follows: \[ an( heta) = \frac{\text{Height}}{\text{Base Distance}}\]Here,

• \( \theta = 30^{\circ} \) or \( 60^{\circ} \).
• This equation rearranges to calculate height: \( h = \text{Base Distance} \times \tan(\theta) \).

This is especially useful when the direct measurement of height is infeasible. By using tangent, we can indirectly measure and compute the various dimensions that make solving trigonometry problems straightforward.

Trigonometric Identity

Trigonometric identities are equations that are true for all angles for which the functions are defined, and they form the backbone of solving more complex trigonometric equations. Basic identities involve the reciprocal, Pythagorean, and co-function identities.

In the given exercise, one such important identity is the correlation among trigonometric functions, like how \[ an(30^{\circ}) = \frac{1}{\sqrt{3}} \]and \[ an(60^{\circ}) = \sqrt{3} \]come into play. These relationships allow us to transform equations and solve for variables when direct measurement is not an option.

• Pythagorean identities relate sine, cosine, and tangent, providing avenues to solve for various parts of a triangle.
• Using angle sum and difference identities can also extend calculations when dealing with non-standard angles.

Understanding these identities unlocks more complex problem-solving strategies, making it easier to compute results in trigonometry.