To determine the height of the telephone pole given the information provided, we can use trigonometric principles. Here’s the step-by-step solution:
1. Given Data:
- Distance from the pole: 36 ft
- Angle of elevation: [tex]\(30^{\circ}\)[/tex]
2. Trigonometric Relationship:
- In a right triangle, the tangent of an angle is the ratio of the opposite side (height of the pole, in this case) to the adjacent side (distance from the pole).
3. Using the Tangent Function:
- [tex]\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)[/tex]
- Here, [tex]\(\theta = 30^{\circ}\)[/tex], the opposite side is the height of the pole (which we are seeking), and the adjacent side is 36 ft.
4. Set Up the Equation:
- [tex]\(\tan(30^{\circ}) = \frac{\text{height}}{36}\)[/tex]
5. Calculate [tex]\(\tan(30^{\circ})\)[/tex]:
- From trigonometry tables or a calculator, we know [tex]\(\tan(30^{\circ}) = \frac{1}{\sqrt{3}}\)[/tex]
6. Solve for the Height:
- [tex]\(\frac{1}{\sqrt{3}} = \frac{\text{height}}{36}\)[/tex]
- Multiply both sides by 36 to isolate the height:
[tex]\[
\text{height} = 36 \times \frac{1}{\sqrt{3}}
\][/tex]
7. Simplify (optional):
- You may leave the height as [tex]\(36 \times \frac{1}{\sqrt{3}}\)[/tex] or simplify it further:
[tex]\[
\text{height} = \frac{36}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{36 \sqrt{3}}{3} = 12 \sqrt{3}
\][/tex]
Thus, the height of the pole is approximately 20.784609690826528 ft, which simplifies to [tex]\(12 \sqrt{3} \, \text{ft}\)[/tex].
Therefore, the height of the pole is [tex]\(12 \sqrt{3} \, \text{ft}\)[/tex].
