Problem 65 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is \(60^{\circ}\) and when he retires 40 meter away from the tree the angle of elevation becomes \(30^{\circ} .\) The breadth of the river is (A) \(20 \mathrm{~m}\) (B) \(30 \mathrm{~m}\) (C) \(40 \mathrm{~m}\) (D) \(60 \mathrm{~m}\)

Step-by-Step Solution

Answer

The breadth of the river is 20 meters, option (A).

Step 1: Understand the Problem

A person is observing the top of a tree standing on the opposite bank of a river. Initially, the angle of elevation is \(60^{\circ}\), and after moving 40 meters away, it becomes \(30^{\circ}\). We need to find the breadth of the river, which is the horizontal distance from the original observation point to the base of the tree.

Step 2: Draw a Diagram

Visualize the problem by drawing a right triangle. The tree, the original point, and the retreat point form two right triangles. The tree's height is the same in both triangles. Label the distance from the original point to the tree as \(x\), and label the height of the tree as \(h\).

Step 3: Use Trigonometric Ratios for First Observation

For the first observation at \(60^{\circ}\), use the tangent ratio: \(\tan(60^{\circ}) = \frac{h}{x}\). Since \(\tan(60^{\circ}) = \sqrt{3}\), we have the equation: \(h = x\sqrt{3}\).

Step 4: Use Trigonometric Ratios for Second Observation

For the second observation at \(30^{\circ}\), use the tangent ratio: \(\tan(30^{\circ}) = \frac{h}{x + 40}\). Since \(\tan(30^{\circ}) = \frac{1}{\sqrt{3}}\), the equation becomes: \(h = \frac{x + 40}{\sqrt{3}}\).

Step 5: Set Equations Equal to Solve for x

Since both expressions equal \(h\), set \(x\sqrt{3} = \frac{x + 40}{\sqrt{3}}\). Multiply through by \(\sqrt{3}\) to clear the fraction: \(3x = x + 40\).

Step 6: Solve for x

Solve the equation \(3x = x + 40\) by first subtracting \(x\) from both sides: \(2x = 40\). Divide by 2 to find \(x = 20\).

Step 7: Conclude with the Breadth of the River

The breadth of the river, which is \(x\), is 20 meters.

Key Concepts

Angle of Elevation

The angle of elevation is a crucial concept in trigonometry, especially when dealing with scenarios involving heights and distances. It refers to the angle formed between the horizontal ground and the line of sight when looking up at an object above that horizontal plane. In our exercise, the angle of elevation changes as the observer moves, which allows us to assess the height and distance relations using two observations.
For example, when initially observing, the angle of elevation is \(60^{\circ}\). This angle suggests a certain steepness when looking at the top of the tree from the bank of the river. When the observer steps back, the angle of elevation decreases to \(30^{\circ}\), indicating that the line of sight becomes less steep as the distance increases. This change is what we utilize in solving for unknown distances or heights.
Understanding angles of elevation helps recognize how height, distance, and observer position interrelate in right-angle triangles, facilitating calculations with trigonometric ratios.

Tangents

Tangents provide a straightforward trigonometric ratio particularly useful in problems involving heights and distances where one side is perpendicular. In our scenario observing the tree, the tangent ratio connects the height of the tree and the base distance between the observer and the tree's bottom.
For the initial observation, using the tangent of \(60^{\circ}\) provides us the relationship \(\tan(60^{\circ}) = \frac{h}{x}\). Knowing that \(\tan(60^{\circ}) = \sqrt{3}\), we can set up the equation \(h = x\sqrt{3}\).
The tangent comes handy again when the observer moves 40 meters back, altering the angle of elevation to \(30^{\circ}\). Now, the tangent relationship is formulated as \(\tan(30^{\circ}) = \frac{h}{x + 40}\), which converts to \(h = \frac{x + 40}{\sqrt{3}}\). Recalling these tangent values makes it easy to express the height in terms of distance, which is key for solving the problem.

Right Triangles

Right triangles form the backbone of trigonometric applications like our river exercise. They consist of a 90-degree angle with two other angles adding up to \(90^{\circ}\). This geometry allows the use of trigonometric ratios to find unknown sides or angles.
In the given problem, two right triangles are formed from two positions of observation: initially at a \(60^{\circ}\) angle and then at \(30^{\circ}\) after moving back. Each triangle shares a common height—both represent the vertical height of the tree.
This setup demonstrates how right triangles allow a systematic approach to break down distances and heights. By labeling known and unknown sides, utilizing trigonometric ratios like the tangent, and recognizing patterns and relationships in these triangles, we solve for the river's breadth. This methodology highlights the practical use of right triangles in real-world trigonometry problems.