To find the height of the telephone pole, given a person standing \(36 \, \text{ft}\) from the base of the pole and an angle of elevation of \(30^\circ\) from the ground to the top of the pole, follow these steps:
1. Understand the problem setup:
- You have a right triangle where:
- The horizontal distance from the person to the pole is the adjacent side.
- The height of the pole is the opposite side.
- The angle of elevation is \(30^\circ\).
2. Use the trigonometric relationship:
- For a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side:
[tex]\[
\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}
\][/tex]
- Here, \(\theta = 30^\circ\), the opposite side is the height of the pole \(h\), and the adjacent side is the distance \(36 \, \text{ft}\).
3. Set up the equation:
[tex]\[
\tan(30^\circ) = \frac{h}{36}
\][/tex]
4. Solve for \(h\):
[tex]\[
h = 36 \times \tan(30^\circ)
\][/tex]
5. Use the known value of \(\tan(30^\circ)\):
- \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)
6. Calculate the height:
[tex]\[
h = 36 \times \frac{1}{\sqrt{3}} = 36 \sqrt{3} \times \frac{1}{3}
\][/tex]
[tex]\[
h = 12 \sqrt{3} \, \text{ft}
\][/tex]
Therefore, the height of the telephone pole is [tex]\(12 \sqrt{3} \, \text{ft}\)[/tex].
