Question

the diagram shows the measurements taken by a surveyor. what is the height of the cliff to the nearest tenth of a metre?

Explanation:

Step1: Find length of QR in right - triangle PQR

In right - triangle $PQR$, $\tan60^{\circ}=\frac{QR}{20}$. Since $\tan60^{\circ}=\sqrt{3}$, we have $QR = 20\sqrt{3}$ m.

Step2: Find height of cliff SR

In right - triangle QRS, $\tan40^{\circ}=\frac{SR}{QR}$. We know $QR = 20\sqrt{3}$ m, so $SR=QR\times\tan40^{\circ}$. Substituting $QR = 20\sqrt{3}$ m, we get $SR = 20\sqrt{3}\times\tan40^{\circ}$.
Now, $\sqrt{3}\approx1.732$, $\tan40^{\circ}\approx0.839$. Then $SR = 20\times1.732\times0.839=20\times1.453148\approx29.1$ m.

Answer:

$29.1$ m