Problem 43 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\mathrm{PQ}\) 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

The height of the tower is \((abc \tan \theta) / (4 \Delta)\), option (A).

Step 1: Understand the setup

We are given that \( PQ \) is a vertical tower and \( P \) is the foot while \( Q \) is the top. Points \( A, B, C \) lie in a horizontal plane through \( P \), and the angles of elevation from these points to \( Q \) are all equal to \( \theta \). The sides of the triangle formed by these points are \( a, b, \text{ and } c \), with an area of \( \Delta \). We need to find the height \( h \) of the tower.

Step 2: Angle of elevation

The angle of elevation \( \theta \) is the angle between the line from a point (like \( A \), \( B \), or \( C \)) to the top of the tower \( Q \) and the horizontal plane. Since these are equal for points \( A, B, \text{ and } C \), the height \( h \) of the tower is the same above the horizontal line drawn from these points.

Step 3: Use trigonometry

In right triangles formed by each point (\( A, B, C \)) and the tower (\( QP \)), the height \( h \) is the opposite side, and the distances from points \( A, B, \text{ and } C \) to the foot of tower \( P \) are the adjacent sides. Therefore, \( \tan \theta = \frac{h}{d} \), where \( d \) is the horizontal distance from any of these points to the foot \( P \).

Step 4: Connect with the given triangle

All three right triangles act like connecting rods of a radius encircling the triangle \( ABC \). If points \( A, B, C \) are joined, the area of the triangle \( ABC \) is \( \Delta \), and the base of any altitude used could be derived using side lengths \( a, b, c \) interconnected such that all affect \( d \).

Step 5: Derive the expression for height

By substituting \( d = \frac{a \times b \times c}{4 \Delta} \) into the trigonometric equation from Step 3, and knowing \( h = d \tan \theta \), we find that the height \( h = \frac{abc \tan \theta}{4 \Delta} \).

Step 6: Check against options

Checking this derived expression against the given options, we see that (A) aligns perfectly, giving us: \( h = \frac{abc \tan \theta}{4 \Delta} \).

Key Concepts

Angle of Elevation

The angle of elevation is an important concept in trigonometry, especially when dealing with heights and distances. It is the angle formed between a horizontal line and the line of sight to an object above the observer. In our problem, the angle of elevation \( \theta \) from the points \( A, B, \text{and} \, C \) to the top of the tower \( Q \) is the same for all these points.

• It represents the steepness of the line of sight.
• The angle is measured upward from the horizontal plane at the observer's point.
• In practical scenarios, identifying this angle helps to determine the height of tall objects from a distance.

Identifying the angle of elevation allows us to use trigonometric ratios, like tangent, to calculate unknown vertical heights. The tangent of the angle of elevation is defined as the ratio of the opposite side (the height of the object) to the adjacent side (the horizontal distance from the observer to the base of the object). In mathematical terms, this is written as \( \tan \theta = \frac{h}{d} \), where \( h \) is height and \( d \) is the horizontal distance.

Right Triangle Properties

Right triangles are fundamental in trigonometry, and understanding their properties is crucial when working with angles and distances. A right triangle has one angle of 90 degrees, making it straightforward to apply trigonometric functions.

• In a right triangle, the side opposite the right angle is called the hypotenuse.
• The other two sides are known as the legs, with one being opposite the angle of interest and the other adjacent to it.
• The Pythagorean theorem applies, allowing us to relate the sides: \( c^2 = a^2 + b^2 \).

In trigonometry, focusing on sides relative to specific angles allows us to define ratios like sine, cosine, and tangent. In this exercise, we particularly use the tangent function, as it connects the angle of elevation to the height of the tower and the horizontal distance, providing a method to discover unknown distances given one angle and side length.

Area of Triangle

The area of a triangle can be calculated in various ways, depending on the data available. When dealing with non-right triangles, Heron's Formula or the basic area formula \( \Delta = \frac{1}{2} \times \text{base} \times \text{height} \) can be used.

• For a triangle with sides \( a, b, \text{and} \, c \), it may be useful to use Heron's formula: \( \Delta = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s \) is the semi-perimeter.
• In the case of a right triangle, the area is easily computed using the base and height.
• The connection between the area and trigonometry is evident through the sine function when using the formula: \( \Delta = \frac{1}{2}ab\sin(C) \).

In this problem, knowing the area \( \Delta \) of triangle \( ABC \) and the lengths of its sides helps us relate the triangle's internal structure to the external setup involving the tower and angles of elevation. In essence, the area formula and the properties of the triangle help compute the required height of the tower as shown by the exercise solution.