Problem 17 - A Complete Resource Book in Mathematics for JEE Main

Question

The angle of elevation of a stationary cloud from a point \(2,500 \mathrm{~m}\) above a lake is \(15^{\circ}\) and the angle of depression of its reflection in the lake is \(45^{\circ}\). The height of the cloud above the lake level is (A) \(2500 \sqrt{3} \mathrm{~m}\) (B) \(2500 \mathrm{~m}\) (C) \(500 \sqrt{3} \mathrm{~m}\) (D) none of these

Step-by-Step Solution

Answer

The height of the cloud above the lake level is approximately 4650 m, which is not one of the given options.

Step 1: Understand the Problem

A person is observing a cloud from a point which is 2500 m above the lake level. The angle of elevation to the cloud is 15°, and the angle of depression to its reflection in the lake is 45°. We are asked to find the height of the cloud above the lake.

Step 2: Analyze Given Information

The point of observation is 2500 m above the lake, i.e., at point P. The cloud is at point C, and its reflection is at point R, directly opposite below the cloud. Given angles are \( \angle CPO = 15^\circ \) and \( \angle PRO = 45^\circ \) with respect to a horizontal line.

Step 3: Use Tan of Angle for Cloud

Using tangent for cloud elevation: \( \tan(15^\circ) = \frac{h - 2500}{x} \), where \( h \) is the height of the cloud above the lake and \( x \) is the horizontal distance from the observer to the point directly below the cloud.

Step 4: Use Tan of Angle for Reflection

Using tangent for reflection: \( \tan(45^\circ) = \frac{h + 2500}{x} \). Since \( \tan(45^\circ) = 1 \), this simplifies to \( h + 2500 = x \).

Step 5: Equate and Solve Equations

From Step 3, \( x = \frac{h - 2500}{\tan(15^\circ)} \) and from Step 4, \( x = h + 2500 \). Set these equal: \( \frac{h - 2500}{\tan(15^\circ)} = h + 2500 \). Solve for \( h \). This leads to: \( h - 2500 = (h + 2500) \cdot \tan(15^\circ) \).

Step 6: Simplify and Calculate

Rearrange: \( h - 2500 = h\tan(15^\circ) + 2500\tan(15^\circ) \). Move terms to get \( h(1 - \tan(15^\circ)) = 2500(1 + \tan(15^\circ)) \). Solve for \( h \): \( h = \frac{2500(1 + \tan(15^\circ))}{1 - \tan(15^\circ)} \). Calculate using \( \tan(15^\circ) \approx 0.2679 \).

Step 7: Solve for Final Answer

Substitute \( \tan(15^\circ) \approx 0.2679 \) into the equation. \( h = \frac{2500(1 + 0.2679)}{1 - 0.2679} \). Simplify to find \( h \): \( h = \frac{2500 \times 1.2679}{0.7321} \). Calculate to get \( h \approx 4650 \text{ m} \).

Key Concepts

Angle of Elevation

The angle of elevation is a concept in trigonometry used to determine the angle between the horizontal and line of sight when looking up at an object. It's measured from a horizontal line to the line of sight upward toward the object.

In our exercise, this angle of elevation is used to find the distance from a point above a lake to a cloud in the sky. We know the angle of elevation is 15° from a point that is 2500 meters above the lake. By using the tangent function, which is central to trigonometry for non-right angle triangles, we can model the problem. This involves creating a right triangle where the angle of elevation, the height of the point above the lake, and the horizontal distance to the base of the cloud form the triangle's components.

To model such a scenario, we use the tangent function:

• The formula becomes: \[tan( heta) = \frac{\text{opposite side}}{\text{adjacent side}} \]Where \(\theta\) is the angle of elevation.
• Using known values, the height difference \((h - 2500)\) becomes the opposite side, and \(x\) the horizontal distance, as the adjacent side.
• Thus, the equation simplifies to \[tan(15^\circ) = \left(\frac{h - 2500}{x}\right). \]

Angle of Depression

The angle of depression is often confusing but it is simply the angle between the horizontal and a line of sight looking down from an elevated position. It’s crucial in contexts where one must determine distances or elevations below the viewer's height.

In the provided exercise, the angle of depression is attached to the observer's sight when looking downwards toward the reflection of a cloud in a lake. That means the observer high above sees the cloud's image at a lower angle, which in this problem is 45°.

The steps include:

• When the observer views the cloud's reflection, the angle below the horizontal is used to form a right triangle.
• Through the tangent function, which is equal to 1 when the angle is 45°, the equation \[tan(45^\circ) = \left(\frac{h + 2500}{x}\right) \]is formulated. With \(x\) representing the horizontal distance.
• Since \(\tan(45^\circ) = 1\), the equation simplifies to \[h + 2500 = x \]with the distance \((h + 2500)\) being equal to the horizontal distance.

Tangent Function

The tangent function is an essential element of trigonometry, frequently used to solve problems involving right triangles and angles of elevation or depression. It’s one of the basic trigonometric functions representing the ratio of two sides in a right triangle:

• Opposite side / Adjacent side

The tangent function is particularly effective when direct measurement of a side isn't feasible, and angle measurement is available.

For trigonometric calculations:

• The formula becomes: \[tan(\theta) = \frac{\text{height/opposite}}{\text{base/adjacent}} \]where \(\theta\) is the known angle.
• In solving the exercise, the tangent function was used twice — once for the cloud above and once for the reflection.
• It helped establish the relationships by leveraging angles 15° and 45°, each providing one equation. Combining these equations by setting the horizontal distances \(x\) equal helped solve for the cloud's height.

Application of the tangent function masterfully links the angles and distances to determine the desired variable, like in this instance, the height of the cloud.