To find the height of the telephone pole, we use trigonometry. The person stands 36 feet away from the base of the pole, and the angle of elevation from the ground to the top of the pole is [tex]\(30^{\circ}\)[/tex].
Here's the step-by-step process:
1. Understand the trigonometric relationship: For this scenario, we are dealing with a right triangle where the angle of elevation [tex]\(\theta\)[/tex] is [tex]\(30^{\circ}\)[/tex], the adjacent side is the distance from the person to the pole (36 feet), and the opposite side is the height of the pole that we need to find.
2. Use the tangent function: The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Here, [tex]\(\tan(30^{\circ}) = \frac{\text{height}}{36}\)[/tex].
3. Solve for the height: Rearrange the equation to solve for the height:
[tex]\[
\text{height} = 36 \times \tan(30^{\circ})
\][/tex]
4. Calculate the tangent of [tex]\(30^{\circ}\)[/tex]: The standard value of [tex]\(\tan(30^{\circ})\)[/tex] is [tex]\(\frac{1}{\sqrt{3}}\)[/tex] or approximately 0.577.
5. Multiply the distance by [tex]\(\tan(30^{\circ})\)[/tex]:
[tex]\[
\text{height} = 36 \times \frac{1}{\sqrt{3}}
\][/tex]
[tex]\[
\text{height} \approx 36 \times 0.577
\][/tex]
6. Compute the final height: Performing the multiplication gives us:
[tex]\[
\text{height} \approx 20.784609690826528
\][/tex]
Thus, the height of the telephone pole is approximately [tex]\(20.784609690826528\)[/tex] feet. However, looking at the options provided in the problem, the nearest mathematically accurate representation of this height is [tex]\(12 \sqrt{3}\)[/tex] feet, since [tex]\( 12\sqrt{3} \approx 20.7846\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{12 \sqrt{3} \text{ ft}} \][/tex]
