To find the height of the telephone pole given the distance from the pole and the angle of elevation, follow these steps:
1. Understand the problem:
You are given:
- The distance from the person to the pole ([tex]\( d = 36 \)[/tex] feet)
- The angle of elevation from the person’s line of sight to the top of the pole ([tex]\( \theta = 30^\circ \)[/tex])
You need to find the height of the pole ([tex]\( h \)[/tex]).
2. Use trigonometric relationships:
Since you know the angle of elevation ([tex]\(\theta\)[/tex]) and the distance from the pole ([tex]\( d \)[/tex]), you can use the tangent function because tangent relates the opposite side (height of the pole, [tex]\( h \)[/tex]) to the adjacent side (distance from the pole, [tex]\( d \)[/tex]) in a right triangle:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
\][/tex]
3. Set up the equation:
Substitute the known values into the tangent formula:
[tex]\[
\tan(30^\circ) = \frac{h}{36 \text{ ft}}
\][/tex]
4. Solve for [tex]\( h \)[/tex]:
[tex]\[
h = 36 \text{ ft} \times \tan(30^\circ)
\][/tex]
5. Recall the value of [tex]\(\tan(30^\circ)\)[/tex]:
[tex]\[
\tan(30^\circ) = \frac{1}{\sqrt{3}}
\][/tex]
6. Perform the calculation:
[tex]\[
h = 36 \text{ ft} \times \frac{1}{\sqrt{3}} \approx 20.784609690826528 \text{ ft}
\][/tex]
The calculated height of the telephone pole is approximately [tex]\( 20.78 \)[/tex] feet. None of the provided options seems to match exactly this value. Therefore, based on the closest matching values and exact trigonometric calculations, the height is not one of the listed options, but the true calculated height is:
[tex]\[ \boxed{20.784609690826528 \text{ ft}} \][/tex]
