To find the height of the telephone pole given that a person is standing 36 feet from the pole and the angle of elevation to the top of the pole is 30 degrees, we will use trigonometric principles.
1. Identify the relevant trigonometric function:
Since we know the distance from the person to the pole (the adjacent side of the right triangle) and we need to find the height of the pole (the opposite side), we use the tangent function. The tangent function relates the angle of a right triangle to the ratio of the opposite side to the adjacent side:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
2. Substitute the known values into the tangent function:
Here, [tex]\(\theta = 30^\circ\)[/tex] and the adjacent side (distance from person to pole) is 36 feet.
[tex]\[
\tan(30^\circ) = \frac{\text{height of the pole}}{36 \text{ ft}}
\][/tex]
3. Solve for the height of the pole:
We rearrange the formula to solve for the height of the pole (opposite side):
[tex]\[
\text{height of the pole} = 36 \text{ ft} \times \tan(30^\circ)
\][/tex]
4. Use the tangent of 30 degrees:
Recall that [tex]\(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)[/tex] or approximately 0.5774.
[tex]\[
\text{height of the pole} = 36 \text{ ft} \times 0.5774
\][/tex]
5. Calculate the height:
Multiplying 36 feet by the tangent of 30 degrees:
[tex]\[
\text{height of the pole} \approx 36 \text{ ft} \times 0.5774 \approx 20.7846 \text{ ft}
\][/tex]
After performing the calculation, we find that the height of the pole is approximately 20.78 feet, which matches none of the provided options exactly. However, it is closest to the value [tex]\(12 \sqrt{3}\)[/tex].
Therefore, the height of the telephone pole is [tex]\(12 \sqrt{3} \text{ ft}\)[/tex], which is approximately equal to 20.7846 feet.
