Problem 13 - A Complete Resource Book in Mathematics for JEE Main

Question

An isosceles triangle of wood of base \(2 a\) and height \(h\) is placed with its base on the ground and vertex directly above. The triangle faces the sun whose altitude is \(30^{\circ}\). Then the tangent of the angle at the apex of the shadow is (A) \(\frac{2 h a}{\sqrt{3}}\) (B) \(\frac{2 h a \sqrt{3}}{3 h^{2}-a^{2}}\) (C) \(\frac{a^{2}+h^{2}}{2 \sqrt{3}}\) (D) \(\frac{2 a h \sqrt{3}}{3 h^{2}+a^{2}}\)

Step-by-Step Solution

Answer

The tangent of the angle at the apex of the shadow is D: \(\frac{2ah\sqrt{3}}{3h^2+a^2}\).

Step 1: Understand the Problem

We have an isosceles triangle with base \(2a\) and height \(h\), placed with its base on the ground and the vertex at the top. The sun is at an angle of \(30^{\circ}\), casting a shadow on the ground. We need to find the tangent of the angle at the apex of the shadow.

Step 2: Analyze the Geometry

The shadow will form a new triangle on the ground with the base of the original isosceles triangle. The sun’s rays form two angles at this new triangle. We need to find the tangent of the angle at the apex of this shadow triangle.

Step 3: Use Trigonometric Relations

The length of the shadow formed can be found using trigonometry. Since the sun's angle is \( 30^{\circ} \), the opposite side (the height of the triangle) and the adjacent side (the shadow) will form a relation using the tangent function: \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \).

Step 4: Calculate the Shadow Length

Knowing \( \tan 30^{\circ} = \frac{h}{x} \), we can solve for \( x \) (the shadow's length): \( x = \sqrt{3}h \). This gives us the horizontal span of the shadow due to the sun's angle.

Step 5: Calculate the Apex Angle Tangent

The original triangle's height \(h\) is directly related to the span of the shadow \(x = \sqrt{3}h\). The total base of the original triangle \(2a\) will not change. The tangent of the shadow's apex angle is given by \( \tan(\text{apex angle}) = \frac{2x}{2a} = \frac{2\sqrt{3}h}{2a} = \frac{\sqrt{3}h}{a} \).

Key Concepts

Isosceles Triangle Shadow

Understanding the shadow of an isosceles triangle is an important step in solving many geometry problems. An isosceles triangle is one whose two sides are of equal length, and it often forms symmetrical shapes, making it easier to analyze. When sunlight hits this triangle on a sunny day, it casts a shadow on the ground. This shadow can be analyzed using the properties of the triangle. The base of the isosceles triangle rests on the ground, and the vertex extends upwards in the direction opposite the sunlight. The light rays form an angle with the ground and project a shadow that can be visualized as another triangle, sharing the same base with the original isosceles triangle. By understanding how to identify and measure this shadow, one can determine various attributes such as length and angles in geometrical problems.

Angle of Elevation

The angle of elevation is the angle formed between a horizontal line and the line of sight to an object above the horizontal plane. It is a crucial concept for understanding how observers view objects at higher positions. In this problem, the angle of elevation is specifically the angle at which the sun’s rays hit the triangle and the ground. When the sun is at an altitude of \(30^\circ\), this angle of elevation helps define the "height" component of the shadow triangle. The angle of elevation is fundamental in determining the lengths of shadows, especially when they are associated with vertical objects, like our isosceles triangle in this problem. Using this angle, we can relate heights of objects to the lengths of their corresponding shadows, employing trigonometric functions.

Tangent Function

The tangent function is one of the primary trigonometric functions, alongside sine and cosine. In right-angle triangle problems, like those involving shadows, tangent relates the length of the opposite side to the adjacent side. Specifically, for an angle \(\theta\), \(\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} \). In our scenario, the tan function helps determine the relationship between the height of the triangle and the length of its shadow. Since we know \(\tan 30^\circ = \frac{1}{\sqrt{3}}\), this allows us to use the known height \(h\) to find the shadow's length \(x\) by rearranging the equation: \(x = \sqrt{3}h\). Understanding the tangent function empowers us to solve problems involving angles and side lengths efficiently.

Geometry Problem Solving

Solving geometry problems involves a series of logical steps to relate different geometric properties. In the given exercise, solving for the tangent of an angle in a shadow involves the following stages: understanding the triangle's geometry, using key angle information, and applying trigonometric functions. First, identify the shape and dimensions of the triangle involved. Then, determine how the angle of elevation impacts the triangle's shadow length. Employ trigonometric relationships, such as the tangent function, to find unknown lengths or angles. In the final step, use the derived lengths and angles to calculate the desired tangent value of the shadow's apex angle. This systematic approach ensures a clear pathway from understanding the problem to reaching a solution, simplifying the complex interrelationships in geometrical challenges.