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A balloon is moving up in air vertically above a point $$A$$ on the ground. When it is at a height $$h_1$$, a girl standing at a distance $$d$$ (point B) from $$A$$ (see figure) sees it at an angle $$45^\circ$$ with respect to the vertical. When the balloon climbs up a further height $$h_2$$, it is seen at an angle $$60^\circ$$ with respect to the vertical if the girl moves further by a distance $$2.464\,d$$ (point C). Then the height $$h_2$$ is (given $$\tan 30^\circ = 0.5774$$):
$$\text{From geometry, angle with horizontal} = 90^\circ - \text{angle with vertical}$$
$$\theta_B = 90^\circ - 45^\circ = 45^\circ$$
$$\theta_C = 90^\circ - 60^\circ = 30^\circ$$
$$\tan(45^\circ) = \frac{h_1}{d} \implies 1 = \frac{h_1}{d} \implies h_1 = d$$
$$\tan(30^\circ) = \frac{h_1 + h_2}{d + 2.464d} = \frac{d + h_2}{3.464d}$$
$$0.5774 = \frac{d + h_2}{3.464d} \implies d + h_2 = 0.5774 \times 3.464d$$
$$d + h_2 \approx 2.000d \implies h_2 = d$$
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