A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain poin

Q: Wait, do I need to know the actual height of the pillar to solve this?

That is a fantastic question! The beauty of this problem lies in the fact that the height $h$ is a common factor in all our expressions. When we set up the ratio of distances to find the time, the $h$ terms will cancel out perfectly. So, no, you do not need the height!

Q: Why can't we just use the sine rule instead of tangent?

You absolutely could use the sine rule, but it would be like using a sledgehammer to crack a nut! Since we are dealing with right-angled triangles, the tangent function is the most direct and elegant tool. It keeps the algebra clean and minimizes the chance of calculation errors.

Q: What if the man's speed wasn't uniform? Would the answer change?

Yes, it would! If the speed were not uniform, we would need a function for the velocity, $v(t)$, and we would have to use calculus to integrate the distance. But in JEE problems, 'uniform speed' is our signal that we can use simple linear kinematics: $\text{Distance} = \text{Speed} \times \text{Time}$.

Question

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30°. After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 60°. Then the time taken (in minutes) by him, from B to reach the pillar, is:

Options:

(A) 20

(B) 5

(C) 6

(D) 10

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a flat, open plain, looking up at a majestic, vertical pillar. You start walking towards it, and at a specific point $A$, you glance up at the top. The angle of elevation is $30^\circ$.

As you continue your steady, uniform walk for ten minutes, you reach a new point $B$. Now, the pillar looms larger, and the angle of elevation has sharpened to $60^\circ$.

### The Trigonometric Foundation

Let the height of the pillar be $h$. Let $P$ be the base of the pillar and $C$ be its top. We have two right-angled triangles: $	riangle APC$ and $	riangle BPC$.

In $	riangle APC$, the angle of elevation is $30^\circ$. Using the definition of the tangent function:

$$	an(30^\circ) = \frac{PC}{AP} = \frac{h}{AP}$$

Since $	an(30^\circ) = \frac{1}{\sqrt{3}}$, we find that the distance $AP = h\sqrt{3}$.

Now, look at the smaller triangle, $	riangle BPC$, where the angle is $60^\circ$. Here:

$$	an(60^\circ) = \frac{PC}{BP} = \frac{h}{BP}$$

Since $	an(60^\circ) = \sqrt{3}$, we rearrange this to find $BP = \frac{h}{\sqrt{3}}$.

### The Kinematic Bridge

We know the man walked from $A$ to $B$ in exactly $10$ minutes. The distance $AB$ is the difference between the two base lengths: $AB = AP - BP$.

Substituting our expressions, we get:

$$AB = h\sqrt{3} - \frac{h}{\sqrt{3}}$$

To simplify this, we find a common denominator:

$$AB = \frac{3h - h}{\sqrt{3}} = \frac{2h}{\sqrt{3}}$$

This is the distance covered in $10$ minutes. Since the speed $v$ is uniform, we calculate it as:

$$v = \frac{	ext{Distance } AB}{	ext{Time}} = \frac{\frac{2h}{\sqrt{3}}}{10} = \frac{h}{5\sqrt{3}}$$

### The Final Stretch

Now, we need to find the time taken to cover the remaining distance $BP$. We know $BP = \frac{h}{\sqrt{3}}$ and our speed $v = \frac{h}{5\sqrt{3}}$.

The time taken is simply:

$$	ext{Time} = \frac{	ext{Distance } BP}{	ext{Speed } v} = \frac{\frac{h}{\sqrt{3}}}{\frac{h}{5\sqrt{3}}}$$

Notice the elegance here: the $h$ cancels out, the $\sqrt{3}$ cancels out, and we are left with:

$$	ext{Time} = 5 	ext{ minutes}$$

It is a beautiful, clean result. The man takes exactly **5 minutes** to reach the pillar from point $B$.

Select Answer:

Visualized Solution

Visualizing the Scenario

Moving to Point B

Distance from A to Pillar

Distance from B to Pillar

Calculating Distance AB

Simplifying Distance AB

Calculating Speed

Setting up Time for BP

Final Calculation

The Sigma Insight: Heights and Distances

Analyzing the Setup

The Trigonometric Foundation

The Kinematic Bridge

The Final Stretch

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