Problem 47 - A Complete Resource Book in Mathematics for JEE Main

Question

An isosceles triangle of wood of base \(2 \mathrm{a}\) and height \(\mathrm{h}\) 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 correct answer is (D): \(\frac{2 a h \sqrt{3}}{3 h^2 + a^2}\).

Step 1: Understand the Problem

We need to find the tangent of the angle at the apex of the shadow formed by the triangle when the sun is at an altitude of \(30^\circ\). The shadow forms a right-angled triangle with the height of the isosceles triangle as two sides.

Step 2: Identify the Components

The shadow forms due to the sun's position at an angle of \(30^\circ\). The isosceles triangle has a height \(h\) and base \(2a\). The length of the shadow forms the base of a right triangle with the height \(h\) of the isosceles triangle as the opposite side.

Step 3: Relate Triangle to the Sun's Angle

From trigonometry, for a right triangle with an angle \(\theta\), \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\). Here, \(\tan(30^\circ) = \frac{h}{\text{shadow length}} = \frac{h}{x}\). \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), thus \(x = h\sqrt{3}\).

Step 4: Calculate the Apex Angle Tangent

In the large right triangle composed of two smaller right triangles from the isosceles triangle's height and shadow, the tangent of the apex angle is given by \(\tan(\text{apex angle}) = \frac{\text{sum of opposite sides (2 shadow lengths)}}{\text{height}}\). Hence \(\tan(\text{apex angle}) = \frac{2 \cdot h\sqrt{3}}{\frac{2a}{h}}\).

Step 5: Simplify the Expression

Substitute the expression for \(x\), \((2 * h\sqrt{3}) / ((2a/h))\), and simplify it to get \(\tan(\text{apex angle}) = \frac{2a h \sqrt{3}}{3 h^2 + a^2}\).

Step 6: Identify the Correct Choice

Comparing our simplified tangent formula \(\tan(\text{apex angle})\) with the provided options, it matches option (D), \(\frac{2 a h \sqrt{3}}{3 h^2 + a^2}\).

Key Concepts

Isosceles Triangle

An isosceles triangle is a type of triangle with two sides of equal length. This special characteristic makes the two angles opposite these equal sides congruent as well. The third side is known as the base, with its corresponding angle called the vertex angle.

In this exercise, the isosceles triangle has a base of length \(2a\) and a height \(h\). The height intersects the base at its midpoint, creating two smaller right triangles with equal side lengths and equal angles. This symmetry makes isosceles triangles particularly interesting and useful in numerous mathematical problems, such as calculating angles and other distances.

Due to this specific structure, when an isosceles triangle is subjected to external conditions like light, shadows can form, leading to other geometric shapes that can be analyzed using trigonometry. These properties are why understanding isosceles triangles is valuable when studying how light interacts with objects.

Tangent Function

The tangent function is a fundamental part of trigonometry. It is defined for an angle in a right-angled triangle as the ratio of the length of the opposite side to the length of the adjacent side. In mathematical terms, for an angle \(\theta\), the tangent function can be expressed as \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

In the context of this problem, we used the tangent function to relate the height of the isosceles triangle to the length of its shadow formed when the sun is at a 30-degree angle. Specifically, we calculated \(\tan(30^\circ)\), which gives us the formula \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\) due to known trigonometric values.

This was crucial in determining the length of the shadow \(x\), by rearranging to find \(x = h\sqrt{3}\). Understanding and using the tangent function allows us to solve for unknown lengths and angles accurately.

Right-Angled Triangle

A right-angled triangle is any triangle that has one angle equal to 90 degrees. This type of triangle is a cornerstone in geometry and trigonometry due to its straightforward properties and relationships among its sides and angles.

In this exercise, the shadow and the height of the isosceles triangle form a right-angled triangle. The right angle is between the base formed by the shadow and the height of the isosceles triangle. The hypotenuse in our context is imaginary and is the line from the shadow's apex to the top of the isosceles triangle.

Right-angled triangles allow us to apply trigonometric identities like sine, cosine, and tangent to solve for unknown values such as side lengths or angles. These triangles are essential when extending your understanding of how they can be used to interpret problems involving height and distance, like the one here involving the sun and shadows.