Question : The length of the shadow of a vertical pole on the ground is 36 m. If the angle of elevation of the sun at that time is $\theta$. Such that $\sec \theta=\frac{13}{12}$, then what is the height (in m) of the pole?
Option 1: 12
Option 2: 15
Option 3: 18
Option 4: 9
Correct Answer: 15
Solution :
The angle of elevation of the sun and the height of the pole form a right triangle with the length of the shadow.
Given that $\sec \theta = \frac{13}{12}$,
$⇒\cos \theta = \frac{1}{\sec \theta} = \frac{12}{13}$
Since $\cos^2 \theta + \sin^2 \theta = 1$.
$⇒\sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(\frac{12}{13}\right)^2} = \frac{5}{13}$
$⇒\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{5}{12}$
Therefore, the height of the pole = $\frac{5}{12} \times 36 \, \text{m} = 15 \, \text{m}$
Hence, the correct answer is 15.
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