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 [tex]h[/tex] meters and [tex]h= h_1+h_s[/tex].
Using trigonometric ratios:
tan (Ф) = opposite/adjacent
In ΔSPQ. In ΔSPR.
⇒ [tex]tan(60) = \frac{h_1}{x}[/tex] ...equation (i) ⇒ [tex]tan (30)=\frac{h_1}{x+50}[/tex] ...equation (ii)
Dividing equation (i) and (ii)
⇒ [tex]\frac{tan(60)}{tan(30)} = \frac{h_1}{x} \times \frac{x+50}{h_1}[/tex]
⇒ [tex]3 = \frac{x+50}{x}[/tex]
⇒ [tex]3x=x+50[/tex]
⇒ [tex]3x-x=50[/tex]
⇒ [tex]2x=50[/tex]
⇒ [tex]x=\frac{50}{2}[/tex]
⇒ [tex]x=25[/tex] meters
To find [tex]h_1[/tex] plugging [tex]x=25[/tex] in equation (i)
⇒ [tex]h_1=x\times tan(60)[/tex]
⇒ [tex]h_1=25\times 1.73[/tex]
⇒ [tex]h_1=43.25[/tex] meters
The height of the cliff from ground :
⇒ [tex]h= h_1+h_s[/tex]
⇒ [tex]h= 43.25+1.8[/tex]
⇒ [tex]h=45.05[/tex] meters
The cliff is 45 m tall to the nearest meter.
