To determine the height of the telephone pole, we can use trigonometry. Specifically, we'll use the tangent function, which relates the angle of elevation ([tex]\(\theta\)[/tex]), the distance from the observation point to the base of the pole (d), and the height of the pole (h).
Given:
- The distance from the person to the pole (d) = 36 feet
- The angle of elevation ([tex]\(\theta\)[/tex]) = [tex]\(30^\circ\)[/tex]
We know from trigonometry that:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
In this context:
- The "opposite" side is the height of the pole (h).
- The "adjacent" side is the distance from the person to the pole (d).
Thus, we have the equation:
[tex]\[ \tan(30^\circ) = \frac{h}{36} \][/tex]
Now, [tex]\(\tan(30^\circ)\)[/tex] is a known value:
[tex]\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \][/tex]
Substituting this into our equation gives:
[tex]\[ \frac{1}{\sqrt{3}} = \frac{h}{36} \][/tex]
To solve for [tex]\(h\)[/tex], we multiply both sides by 36:
[tex]\[ h = 36 \times \frac{1}{\sqrt{3}} \][/tex]
To rationalize the denominator:
[tex]\[ h = 36 \times \frac{1}{\sqrt{3}} = 36 \times \frac{\sqrt{3}}{3} = 36 \div 3 \times \sqrt{3} = 12\sqrt{3} \][/tex]
Now, verifying the exact numeric result, the height of the pole is approximately 20.784609690826528 feet.
Therefore, the correct option that matches this height in its simplest form is:
[tex]\[ \boxed{12\sqrt{3}\text{ ft}} \][/tex]
