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

Let us denote

$$h$$ = height of the pillar,

$$x$$ = horizontal distance of point $$A$$ from the foot of the pillar,

$$v$$ = uniform speed of the man (in units of distance per minute).

At point $$A$$ the angle of elevation of the top of the pillar is $$30^\circ$$. The right-angled triangle formed has the pillar as the opposite side and the distance $$x$$ as the adjacent side. We therefore write the tangent relation:

$$\tan 30^\circ=\frac{\text{opposite}}{\text{adjacent}}=\frac{h}{x}.$$

We know the standard value $$\tan 30^\circ=\frac1{\sqrt3}$$, so

$$\frac1{\sqrt3}=\frac{h}{x}\quad\Longrightarrow\quad h=\frac{x}{\sqrt3}.$$

Now the man walks for 10 minutes from $$A$$ to $$B$$ at speed $$v$$. Hence the horizontal distance he covers is

$$AB = 10v.$$

The new horizontal distance of point $$B$$ from the pillar’s foot is therefore

$$x-AB=x-10v.$$

At $$B$$ the angle of elevation becomes $$60^\circ$$, so again using the tangent formula,

$$\tan 60^\circ=\frac{h}{x-10v}.$$

The standard value $$\tan 60^\circ=\sqrt3$$ gives

$$\sqrt3=\frac{h}{x-10v}\quad\Longrightarrow\quad h=\sqrt3\,(x-10v).$$

But we already found $$h=\dfrac{x}{\sqrt3}$$. Equating the two expressions for $$h$$:

$$\frac{x}{\sqrt3}=\sqrt3\,(x-10v).$$

Multiply both sides by $$\sqrt3$$ to clear the denominator:

$$x = 3x - 30v.$$

Bring like terms together:

$$x - 3x = -30v \quad\Longrightarrow\quad -2x = -30v \quad\Longrightarrow\quad 2x = 30v.$$

Divide by $$2$$:

$$x = 15v.$$

The remaining horizontal distance from $$B$$ to the pillar is

$$x-10v = 15v - 10v = 5v.$$

Time is distance divided by speed, so the time required from $$B$$ to reach the pillar is

$$\text{time}=\frac{5v}{v}=5\text{ minutes}.$$

Hence, the correct answer is Option B.