1. Set up the geometry

Let the balloon be vertically above point $A$.

• Girl is initially at point $B$, with $AB = d$.
• At that instant, balloon height is $h_1$.
• The girl sees the balloon at an angle $45^\circ$ with respect to the vertical.

Since angle is with the vertical, the angle with the horizontal is: $90^\circ - 45^\circ = 45^\circ$

So, $\tan 45^\circ = \frac{h_1}{d} = 1$ $\Rightarrow h_1 = d$

2. Second observation

The girl moves further from $B$ to $C$ by a distance $2.464d$. So total horizontal distance from $A$ becomes: $AC = AB + BC = d + 2.464d = 3.464d$

Now the balloon has risen by additional height $h_2$, so its new height is: $h_1 + h_2$

It is now seen at an angle $60^\circ$ with respect to the vertical. Thus angle with horizontal is: $90^\circ - 60^\circ = 30^\circ$

Hence, $\tan 30^\circ = \frac{h_1 + h_2}{AC}$

Given: $\tan 30^\circ = 0.5774$ So, $0.5774 = \frac{h_1 + h_2}{3.464d}$

Therefore, $h_1 + h_2 = 0.5774 \times 3.464d$

Now, $0.5774 \times 3.464 \approx 2.0001 \approx 2$

So, $h_1 + h_2 = 2d$

Since $h_1 = d$, $d + h_2 = 2d$ $\Rightarrow h_2 = d$

3. Check options

• A: $0.464d$ ❌
• B: $d$ ✅
• C: $0.732d$ ❌
• D: $1.464d$ ❌

Thus the correct option is: $\boxed{\text{B: } d}$