To determine the height of the telephone pole, we need to use the information given about the distance from the person to the pole and the angle of elevation from the ground to the top of the pole.
Here are the given values:
- Distance from the person to the pole (adjacent side of the right triangle): 36 feet
- Angle of elevation: 30 degrees
We need to find the height of the pole (opposite side of the right triangle).
To solve this, we can use the tangent trigonometric function. In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side:
[tex]\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Given:
[tex]\[
\tan(30^\circ) = \frac{\text{height of the pole}}{36\ \text{feet}}
\][/tex]
We know from trigonometric tables or calculators that:
[tex]\[
\tan(30^\circ) = \frac{1}{\sqrt{3}}
\][/tex]
Substituting the known values into the equation:
[tex]\[
\frac{1}{\sqrt{3}} = \frac{\text{height of the pole}}{36\ \text{feet}}
\][/tex]
Solving for the height of the pole:
[tex]\[
\text{height of the pole} = 36\ \text{feet} \times \frac{1}{\sqrt{3}}
\][/tex]
Simplifying this expression:
[tex]\[
\text{height of the pole} = \frac{36}{\sqrt{3}}
\][/tex]
Rationalizing the denominator:
[tex]\[
\text{height of the pole} = \frac{36}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{36 \sqrt{3}}{3} = 12 \sqrt{3}\ \text{feet}
\][/tex]
Therefore, the height of the pole is:
[tex]\[
\boxed{12 \sqrt{3} \text{ ft}}
\][/tex]
