From the top of a church spire 96 m high, the angles of depression of two vehicles on a road, at the same level as the base of the spire and on the same side of it are x° and y°, where tan x° = $\Big(\dfrac{1}{4}\Big)$ and tan y° = $\Big(\dfrac{1}{7}\Big)$. Calculate the distance between the vehicles.

Let AB be the height of church = 96 m.

Hence, the angle of elevation of the first vehicle at position D to the top of the church is x° and that of the second vehicle at position C is y°.

Considering right angled △ABD, we get

$\Rightarrow \tan x^{\circ} = \dfrac{\text{perpendicular}}{\text{base}} = \dfrac{AB}{BD} \\[1em] \Rightarrow \dfrac{1}{4} = \dfrac{96}{BD} \\[1em] \Rightarrow BD = 96 \times 4 \\[1em] \Rightarrow BD = 384 \text{ m.}$

Considering right angled △ABC, we get

$\Rightarrow \tan y^{\circ} = \dfrac{\text{perpendicular}}{\text{base}} = \dfrac{AB}{BC} \\[1em] \Rightarrow \dfrac{1}{7} = \dfrac{96}{BC} \\[1em] \Rightarrow BC = 96 \times 7 \\[1em] \Rightarrow BC = 672 \text{ m.}$

The distance between the vehicles = 672 m - 384 m = 288 m.

Hence, distance between the vehicles is 288 m.