The angle of elevation of the top $$P$$ of a tower from the feet of one person standing due south of the tower is 45$$^\circ$$ and from the feet of another person standing due west of the tower is 30$$^\circ$$. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to

The height of the tower is $$h = 5$$ m. Person 1 stands due south with angle of elevation $$45°$$, and person 2 stands due west with angle of elevation $$30°$$.

$$ \tan 45° = \frac{h}{d_1} \Rightarrow 1 = \frac{5}{d_1} \Rightarrow d_1 = 5 \text{ m} $$

$$ \tan 30° = \frac{h}{d_2} \Rightarrow \frac{1}{\sqrt{3}} = \frac{5}{d_2} \Rightarrow d_2 = 5\sqrt{3} \text{ m} $$

Since south and west are perpendicular directions, the two persons and the base of the tower form a right triangle.

$$ D = \sqrt{d_1^2 + d_2^2} = \sqrt{25 + 75} = \sqrt{100} = 10 \text{ m} $$

The distance between the two persons is Option B: 10 m.