Can we find the distance of the tree from the road?

Can we find the distance of the tree from the road?

Question

A road runs from $A$ due East to $B$, $500$ yd. distant. The bearings of a tree from $A$ and $B$ are $048^\circ$ and $296^\circ$ respectively. Find the distance, correct to the nearest yard, of the tree from the road.

Solution

In our diagram, $h$ represents the perpendicular distance of the tree from the road and $x$ is the distance from the foot of the perpendicular to $A$.

From the left hand triangle, we find that

$$\begin{equation} \tan 42^\circ=\frac{h}{x}, \end{equation}$$

and from the right hand triangle,

$$\begin{equation} \tan 26^\circ=\frac{h}{500-x}. \end{equation}$$

Rearranging equation $(1)$, we find that $x=\dfrac{h}{\tan 42^\circ}$. Substituting this into $(2)$, we find that $\tan 26^\circ=\frac{h}{500-\dfrac{h}{\tan 42^\circ}}.$ Rearranging, we find that $h\left(1+\frac{\tan 26^\circ}{\tan 42^\circ}\right)=500\tan 26^\circ,$ and so $h=\frac{500\tan 26^\circ}{1+\dfrac{\tan26^\circ}{\tan42^\circ}}.$

This gives us that $h=158$ yd. to the nearest yard.