Question

Sarah’s stands on a ground and sights the top of a steep Clift at a 60 degree angle of elevation she then steps back 50 meters then sights the top of the steep cliff at a 30 degree angle if Sarah is 1.8 meters tall how tall is the cliff to the nearest meter ?

Answer 1

Answer:

Cliff is 45 m tall.

Step-by-step explanation:

Given:

Height of Sarah = 1.8 m

Angle of elevation = 60°

Angle of elevation 50 m back = 30°

As shown in the figure we have two right angled triangles SPQ and SPR.

Let the height of the cliff be $h$ meters and $h= h_1+h_s$.

Using trigonometric ratios:

tan (Ф) = opposite/adjacent

In ΔSPQ.                                        In ΔSPR.

⇒ $tan(60) = \frac{h_1}{x}$ ...equation (i)        ⇒ $tan (30)=\frac{h_1}{x+50}$  ...equation (ii)

Dividing equation (i) and (ii)

⇒ $\frac{tan(60)}{tan(30)} = \frac{h_1}{x} \times \frac{x+50}{h_1}$

⇒ $3 = \frac{x+50}{x}$

⇒ $3x=x+50$

⇒ $3x-x=50$

⇒ $2x=50$

⇒ $x=\frac{50}{2}$

⇒ $x=25$ meters

To find $h_1$ plugging $x=25$ in equation (i)

⇒ $h_1=x\times tan(60)$

⇒ $h_1=25\times 1.73$

⇒ $h_1=43.25$ meters

The height of the cliff from ground :

⇒ $h= h_1+h_s$

⇒ $h= 43.25+1.8$

⇒ $h=45.05$ meters

The cliff is 45 m tall to the nearest meter.