Question : The length of the shadow of a vertical pole on the ground is 18 m. If the angle of elevation of the sun at that time is $\theta$, such that $\cos \theta=\frac{12}{13}$, then what is the height (in m) of the pole?
Option 1: 7.5
Option 2: 9
Option 3: 18
Option 4: 12
Correct Answer: 7.5
Solution :
Let the height of the pole be h and the length of the shadow of the pole = base of the triangle = 18 m
In $\triangle$ABC,
⇒ $\cos \theta =\frac{\text{Base}}{\text{Hypotenuse}}$ = $\frac{12}{13}$
⇒ $\frac{18}{\text{Hypotenuse}}$ = $\frac{12}{13}$
$\therefore$ Hypotenuse = $\frac{18×13}{12}$ = $\frac{39}{2}$ m
Now,
⇒ ($\frac{39}{2}$)
2
= (18)
2
+ (h)
2
⇒ h
2
= $\frac{1521}{4}$ - 324
⇒ h
2
= $\frac{1521 - 324×4}{4}$
⇒ h
2
= $\frac{225}{4}$
$\therefore$ h
= $\frac{15}{2}$ = 7.5 m
Hence, the correct answer is 7.5 m.
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