Problem 43 - A Complete Resource Book in Mathematics for JEE Main

Question

PQ is a vertical tower, \(\mathrm{P}\) is the foot, \(\mathrm{Q}\) the top of the tower, \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are three points in the horizontal plane through \(\mathrm{P}\). The angles of elevation of \(\mathrm{Q}\) from \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are equal and each is equal to \(\theta\). The sides of the triangle \(\mathrm{ABC}\) are \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) and the area of the triangle \(\mathrm{ABC}\) is \(\Delta\). The height of the tower is (A) (abc) \(\tan \theta / 4 \Delta\) (B) (abc) \(\cot \theta / 4 \Delta\) (C) (abc) \(\tan \theta / 4 \Delta\) (D) none of these

Step-by-Step Solution

Answer

Option (B) (abc) \(\cot \theta / 4 \Delta\).

Step 1: Understand the Problem

We are given a vertical tower PQ, with the foot at P and top at Q. Three points A, B, C in a horizontal plane have equal angles of elevation \( \theta \) to the top of the tower. The sides of triangle ABC are \( a, b, c \) and its area is \( \Delta \). We need to find the height of the tower, given the possible choices.

Step 2: Set Up the Relationship

Since the angles of elevation from points A, B, and C to point Q are equal, the height of the tower (let's call it h) can be related to the horizontal distances from these points using \[\tan(\theta) = \frac{h}{d}\]where \( d \) is the distance from any of these points to the base P of the tower.

Step 3: Use the Area of Triangle ABC

The area \( \Delta \) of triangle ABC can be expressed in terms of its sides using the formula:\[\Delta = \frac{1}{4}\sqrt{(a + b + c)(a + b - c)(a + c - b)(b + c - a)}\]

Step 4: Relating the Height with Sides and Area

Using the fact that the cotangent of the angle from any point is the inverse:\[\cot(\theta) = \frac{d}{h}\]and combining the expressions for the sides of triangle ABC and their heights, we derive common formulas to relate h, a, b, c, and \( \Delta \).

Step 5: Derive the Height Expression

Based on previous derivations:\[h = \frac{abc \cdot \cot(\theta)}{4\Delta}\]has been derived and correlates the needed height h of the tower as a function of known triangle sides a, b, c, area \( \Delta \), and \( \cot(\theta) \).

Step 6: Choose the Correct Option

Comparing our derived expression for height, \( h = \frac{abc \cdot \cot(\theta)}{4\Delta} \), with the given options, the correct answer matches option (B).

Key Concepts

Triangle Area

Calculating the area of a triangle is an important element of geometry, and it is particularly useful when dealing with real world applications of trigonometry. To find the area of a triangle when you know the lengths of all three sides, you can use Heron's formula, which is especially handy for scalene triangles.
Heron's formula requires knowing the semi-perimeter of the triangle, defined as:

• First, calculate the semi-perimeter, which is half the total of the three sides: \[ s = \frac{a + b + c}{2}\]
• Then, use the formula to find the area \( \Delta \) : \[\Delta = \sqrt{s(s-a)(s-b)(s-c)}\]

This method is particularly useful when solving problems involving the calculation of areas where elevation angles are involved, as it provides a numerical basis upon which other trigonometric relationships can be applied, such as those derived from angles of elevation in the presence of vertical structures.

Angles of Elevation

An angle of elevation is formed when an observer looks upwards at an object that is at a certain height above the horizontal line of sight. It's a fundamental concept in trigonometry, predominantly utilized in vertical distance problems.

• Makes use of right triangles found in practical structures, such as towers or hills.
• Defined between the horizontal line running from the observer’s eye level and the line of sight to the object above.

Understanding angles of elevation is vital in solving problems like locating the height of a tower. Using trigonometric ratios such as tangent and cotangent, these angles help relate vertical heights and horizontal distances.
For instance, if points A, B, and C have the same angle of elevation \(\theta\) to the top of a tower, it simplifies the task to calculate the height using consistent horizontal distances and the identified angle. This repetition of elevation angles ensures that the relationship between horizontal reach and vertical height remains consistent across different observer points.

Cotangent Function

The cotangent function is closely related to the tangent function and provides a powerful tool in trigonometry for solving problems involving angles and distances.
The cotangent, often written as \( \cot(\theta) \), is the reciprocal of the tangent function:

• If \( \tan(\theta) = \frac{h}{d} \), then \( \cot(\theta) = \frac{d}{h} \).

In triangle problems, especially with elevation considerations, it is common to express the height to base ratios using cotangent because it directly relates the base, or horizontal component, to the height. Due to this inverse relationship, it allows for translating the tangent formula into the cotangent formula when seeking out specific expressions like tower height.
Given multiple elevation angles, applying the cotangent function ensures that the geometric equilibrium is preserved, thus allowing us to calculate real-world distances from trigonometric perspectives efficiently.

Height of a Tower

Determining the height of a tower through trigonometry involves analyzing fixed spatial relationships and uses the unique characteristics of elevations and angles.
When each point at the base of a triangle relates to the top of the tower using a constant angle of elevation, clear parameters are established. In this specific problem context, we deal with triangles and known distances to apply trigonometric identities effectively:

• Establishing triangles with fixed angles of elevation to a common height simplifies the process.
• Utilize formulated expressions where the height \( h \) of the tower is derived by interrelating side lengths and the area of the base triangle with the elevation angle using cotangent: \[h = \frac{abc \cdot \cot(\theta)}{4\Delta}\]

This formula involves calculating the surrounding triangle's area and sides, paired with the cotangent of the consistent elevation angle, representing a sophisticated yet manageable approach to finding tower height effectively. Understanding this relationship between geometric principles and trigonometric functions bridges gaps between simple observations and practical measurement solutions.