Problem 14 - A Complete Resource Book in Mathematics for JEE Main

Question

The length of the shadow of a rod inclined at \(10^{\circ}\) to the vertical towards the sun is \(2.05\) metre when the elevation of the sun is \(38^{\circ} .\) The length of the rod is (A) \(\frac{2.05 \sin 38^{\prime \prime}}{\sin 42^{\prime \prime}}\) (B) \(\frac{2.05 \cos 38^{\prime \prime}}{\sin 42^{\prime \prime}}\) (C) \(\frac{2.05 \sin 42^{\prime \prime}}{\sin 38^{\prime \prime}}\) (D) \(\frac{2.05 \cos 42 "}{\sin 38^{n}}\)

Step-by-Step Solution

Answer

The length of the rod is given by option (A).

Step 1: Understanding the problem

We need to find the length of a rod which is inclined at an angle of \(10^{\circ}\) to the vertical. The rod's shadow is \(2.05\) meters long, and the angle of the sun's elevation is \(38^{\circ}\). We are given multiple-choice options for the formula to calculate the rod's length.

Step 2: Analyzing the geometry

Visualize the rod, its shadow on the ground, and the angle of elevation. The rod forms a right triangle with the ground and the shadow. The sun forms an angle of \(38^{\circ}\) with the ground, which is complementary to the angle made by the rod's shadow and the ray of sunlight towards the base of the rod.

Step 3: Using trigonometry to find the rod

Since the rod is inclined \(10^{\circ}\) from the vertical, it forms an effective angle of \(90^{\circ} - 10^{\circ} = 80^{\circ}\) with the horizontal. However, the elevation angle is \(38^{\circ}\), so the angle between the rod and the direction of sunlight beams is \(38^{\circ} + 10^{\circ} = 48^{\circ}\).

Step 4: Applying the sine rule in the triangle

Use the sine rule for a triangle: \[ \frac{\text{side opposite to angle}}{\sin(\text{angle})} = \frac{\text{rod length}}{\sin(42^{\circ})} \]The shadow (2.05 meters) is opposite the angle \(42^{\circ} ( = 90^{\circ} - 38^{\circ})\), therefore:\[ \text{Rod length} = \frac{2.05 \times \sin(38^{\circ})}{\sin(42^{\circ})} \]

Step 5: Matching with given options

Compare the calculated formula to the options given. Option (A): \(\frac{2.05 \sin 38^{\circ}}{\sin 42^{\circ}}\) matches the derived formula for the rod length.

Key Concepts

Sine Rule

Let's delve into the Sine Rule, which is a powerful trigonometric tool used to solve problems involving non-right triangles. The Sine Rule states that in any triangle, the ratio of the length of each side to the sine of its opposite angle is constant. This can be expressed as: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] where \(a\), \(b\), and \(c\) are the sides of the triangle, and \(A\), \(B\), and \(C\) are the angles opposite these sides.
Using the Sine Rule, you can find unknown angles or lengths in triangles when you know some other combination of angles and sides. In our problem involving the rod and the shadow, the Sine Rule helps us find the length of the rod by relating the sides of the triangle formed by the rod, its shadow, and the sunlight using the angles given.

Angle of Elevation

The angle of elevation is a critical concept in many trigonometry problems. The angle of elevation of an object as seen from a certain point is the angle between the line of sight to the object and the horizontal plane. Imagine standing at a plain ground and looking at the sun. The line from your eyes to the sun is the line of sight.
In our context, the problem uses a scenario where the angle of elevation of the sun is given as \(38^{\circ}\). This means that the sun is viewed at an angle of \(38^{\circ}\) above the horizontal level.

• The angle of elevation helps to determine the height of objects using basic trigonometric formulas.
• It is often used with the concepts of right triangle relationships and trigonometric ratios.

Knowing the angle of elevation allows us to apply concepts such as the Sine Rule when incorporating other angles, such as the inclination of the rod in this problem.

Shadow Length Calculation

Calculating the length of a shadow involves understanding the geometry and trigonometry of the situation. Here, the shadow forms the base of a right triangle.
Let's look at some of the fundamental elements you need to consider:

• The shadow length is directly opposite to the angle of elevation in the horizontal plane.
• You need to know or calculate the key angles, as these will allow you to apply the correct trigonometric rules.

In our specific problem, the shadow length is given, which is \(2.05\) meters. By applying the Sine Rule, and knowing the inclination of the rod and the angle of elevation of the sun, we can use these values to establish the relationship between the shadow length, sun elevation, and the angle at the ground. This gives us the equation: \[ \text{Rod length} = \frac{2.05 \times \sin(38^{\circ})}{\sin(42^{\circ})} \] where the angle \(42^{\circ}\) is derived as a complementary angle to the sun's elevation. This relationship helps us determine the actual length of the rod.