To solve this problem, we will make use of trigonometric functions, specifically the tangent function, which relates the angle of elevation to the height of the pole and the distance from the pole.
Given:
- Distance from the person to the pole: [tex]\( 36 \, \text{ft} \)[/tex]
- Angle of elevation to the top of the pole: [tex]\( 30^\circ \)[/tex]
We need to find the height of the pole.
Step-by-step solution:
1. Recall the tangent function in a right triangle:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Here, the angle [tex]\(\theta\)[/tex] is the angle of elevation, the "opposite" side is the height of the pole (which we need to find), and the "adjacent" side is the distance from the person to the pole.
2. Plug in the known values:
[tex]\[
\tan(30^\circ) = \frac{\text{height of the pole}}{36 \, \text{ft}}
\][/tex]
3. Solve for the height of the pole. First, we need the value of [tex]\(\tan(30^\circ)\)[/tex]. From trigonometric tables or a calculator, we know:
[tex]\[
\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.57735
\][/tex]
4. Use this in the equation:
[tex]\[
0.57735 = \frac{\text{height of the pole}}{36 \, \text{ft}}
\][/tex]
5. Multiply both sides by 36 feet to isolate the height:
[tex]\[
\text{height of the pole} = 0.57735 \times 36 \, \text{ft}
\][/tex]
[tex]\[
\text{height of the pole} \approx 20.7846 \, \text{ft}
\][/tex]
Given this calculation, the height of the telephone pole is approximately [tex]\( 20.7846 \, \text{ft} \)[/tex].
None of the provided choices match the calculated height exactly. Therefore, the height of the pole is most accurately expressed as [tex]\( 20.7846 \, \text{ft} \)[/tex].
