A person standing on the bank of a river observes that the angle of elevation of the top of a tree o

Q: Why can't we just use the sine rule? It seems like it would be faster.

You absolutely can! The sine rule is a powerful tool. However, in right-angled triangles, the tangent ratio is often more direct because it relates the two legs directly without needing the hypotenuse. Using $\tan \theta$ keeps the algebra clean and avoids calculating the hypotenuse length, which isn't required here.

Q: Wait, why is the base of the second triangle $x + 40$? Shouldn't it be just $40$?

That is a very common trap! Remember, the person moves $40$ meters away from the river bank. The total distance from the tree is the original breadth of the river ($x$) plus the distance they walked back ($40$). Always draw the full base length from the tree to the observer.

Q: Do I need to worry about the height of the person observing?

In standard JEE problems of this type, unless the height of the observer is explicitly given, we assume the observation is made from ground level. If it were given, you would subtract the observer's height from the total height of the tree to get the effective height for the triangle.

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 meters away from the tree the angle of elevation becomes $30^\circ$. The breadth of the river is

Options:

(A) $60	ext{ m}$

(B) $30	ext{ m}$

(C) $40	ext{ m}$

(D) $20	ext{ m}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the bank of a river, looking across at a majestic tree on the opposite side. We are tasked with finding the breadth of the river, which we will call $x$, and the height of the tree, which we will call $h$.

To solve this, we must translate the physical world into the language of mathematics.

### The First Observation

As you stand at the edge of the river, you look up at the top of the tree. The angle of elevation is $60^{\circ}$. This creates a right-angled triangle where the height of the tree is the opposite side and the breadth of the river is the adjacent side.

Using the tangent ratio, we have:

$$	an 60^{\circ} = \frac{h}{x}$$

Since $	an 60^{\circ} = \sqrt{3}$, we can express the height as $h = x\sqrt{3}$. This is our first crucial link.

### The Second Observation

Now, imagine you walk $40$ meters directly away from the river. You are now at a new point, and the angle of elevation to the top of the tree has decreased to $30^{\circ}$.

The new base of our triangle is the original breadth $x$ plus the $40$ meters you walked, giving us a total base of $x + 40$. Applying the tangent ratio again, we get:

$$	an 30^{\circ} = \frac{h}{x + 40}$$

Since $	an 30^{\circ} = \frac{1}{\sqrt{3}}$, our equation becomes:

$$\frac{1}{\sqrt{3}} = \frac{h}{x + 40}$$

### The Algebraic Bridge

We now have two equations: $h = x\sqrt{3}$ and $\frac{1}{\sqrt{3}} = \frac{h}{x + 40}$. The beauty of this problem lies in the substitution.

By replacing $h$ in the second equation with $x\sqrt{3}$, we eliminate the height entirely:

$$\frac{1}{\sqrt{3}} = \frac{x\sqrt{3}}{x + 40}$$

Now, we cross-multiply to solve for $x$. This gives us:

$$x + 40 = (x\sqrt{3})(\sqrt{3})$$

Simplifying the right side, $\sqrt{3} 	imes \sqrt{3}$ is simply $3$, so we have $x + 40 = 3x$. Subtracting $x$ from both sides, we get $2x = 40$, which leads us to **$x = 20	ext{ m}$**.

The breadth of the river is exactly **$20$ meters**. This problem teaches us that even when we have two unknowns, we can bridge them using the common height of the tree, turning a complex scenario into a simple, elegant algebraic solution.

Select Answer:

Visualized Solution

Visualizing the River and Tree

First Observation at $6 0^{\circ}$

Applying Trigonometry: $tan 6 0^{\circ}$

Finding $h$ in terms of $x$

Moving $40 m$ Away

Second Observation at $3 0^{\circ}$

Applying Trigonometry: $tan 3 0^{\circ}$

Substituting Values

Linking the Equations

Cross-Multiplication

Simplifying the Equation

Solving for $x$

Final Result

The Sigma Insight: Heights and Distances

Analyzing the Setup

The First Observation

The Second Observation

The Algebraic Bridge

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