Problem 27 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A vertical lamp-post of height 9 metres stands at the corner of a rectangular field. The angle of elevation of its top from the farthest corner is \(30^{\circ}\), while from another corner it is \(45^{\circ}\). The area of the field is (A) \(81 \sqrt{2}\) sq.m (B) \(9 \sqrt{2}\) sq.m (C) 9 sq. \(\mathrm{m}\) (D) none of these

Step-by-Step Solution

Answer

The area of the field is \( 81\sqrt{2} \) sq.m, which is option (A).

Step 1: Analyze the Problem

We are given a rectangular field with a vertical lamp-post of height 9 meters. The angles of elevation from the farthest corner and another corner are given as 30° and 45° respectively. We need to find the area of the field.

Step 2: Set Up Equations for Angles of Elevation

Let's denote the two corners as A (where the angle of elevation is 45°) and B (the farthest corner where the angle is 30°). The height of the lamp-post is 9 meters.

Step 3: Calculate Distance for 45° Angle

Using the 45° angle of elevation at corner A, we use the tangent function: \[ \tan(45^{\circ}) = \frac{9}{x} \]Since \( \tan(45^{\circ}) = 1 \), we have \( x = 9 \) meters, where \( x \) is the distance from corner A to the lamp-post.

Step 4: Calculate Distance for 30° Angle

Using the 30° angle of elevation at corner B, we use the tangent function: \[ \tan(30^{\circ}) = \frac{9}{d} \]Since \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \), we get \( d = 9\sqrt{3} \) meters, where \( d \) is the diagonal distance from corner B to the lamp-post.

Step 5: Relate Diagonal with Rectangular Dimensions

In a rectangle, if the diagonal and one side are known, we can use the Pythagorean theorem to find the other side. Denote the two sides of the rectangle as \( a \) (along from A to the lamp-post) and \( b \). Since \( a = x = 9 \), apply Pythagorean theorem:\[ a^2 + b^2 = d^2 \]\[ 9^2 + b^2 = (9\sqrt{3})^2 \]\[ 81 + b^2 = 243 \]\[ b^2 = 162 \rightarrow b = \sqrt{162} = 9\sqrt{2} \]

Step 6: Calculate Area of the Rectangle

The area of the rectangle \( A \) is given by \( a \times b \):\[ A = 9 \times 9\sqrt{2} = 81\sqrt{2} \text{ square meters} \]

Step 7: Conclude with the Correct Option

After calculating the area, we find it matches option (A). Thus, the area of the field is \( 81\sqrt{2} \) square meters.

Key Concepts

Angle of Elevation

The angle of elevation is an important concept in geometry problems, especially in trigonometry. It refers to the angle between the horizontal line from an observer's eye to an object above the eye level. Imagine standing at a corner of a field looking up at the top of a lamp-post; the angle formed between your sight to the top of the post and the line parallel to the ground is the angle of elevation.

In this exercise, we are given two angles of elevation; one of 30° and another of 45° from different corners of the rectangle. These angles help us calculate distances from the corners to the lamp-post by using trigonometric functions such as tangent. Remember:

• A higher angle of elevation means the viewer is closer to the object.
• A lower angle signifies a further distance.

Understanding this concept is crucial as it allows us to use basic trigonometry to find unknown distances or heights.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. In real-world applications, trigonometry helps solve problems by using functions such as sine, cosine, and tangent.

In this problem, trigonometry is used to determine distances from the angles of elevation.Starting with the tangent function, defined as the ratio of the opposite side to the adjacent side of a right triangle, we can find unknown distances. For example:

• With an angle of 45°, since \( \tan(45^{\circ}) = 1 \), the distance from the lamp-post is the same as its height, i.e., 9 meters.
• With an angle of 30°, \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \), so the diagonal distance \( d = 9\sqrt{3} \) meters.

By harnessing the power of trigonometry, you can solve complex geometry problems efficiently.

Pythagorean Theorem

The Pythagorean Theorem is fundamental in geometry. It provides a way to relate the sides of a right-angled triangle. The theorem states: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This is expressed as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.

This principle is applied in our problem to find one side when we already have the length of one side and the diagonal (hypotenuse). Here:

• One side \( a = 9 \) (from the 45° corner).
• The diagonal, calculated using the 30° elevation, is \( 9\sqrt{3} \).
• Substituting these into the theorem, \( 81 + b^2 = 243 \) leads to \( b = 9\sqrt{2} \).

Armed with the Pythagorean Theorem, the area of the rectangular field is then computed as \( 81\sqrt{2} \) square meters.