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A person is standing exactly [tex]36 \, \text{ft}[/tex] from a telephone pole. There is a [tex]30^\circ[/tex] angle of elevation from the ground to the top of the pole.

What is the height of the pole?

A. [tex]12 \, \text{ft}[/tex]
B. [tex]12 \sqrt{3} \, \text{ft}[/tex]
C. [tex]18 \, \text{ft}[/tex]
D. [tex]18 \sqrt{2} \, \text{ft}[/tex]


Sagot :

Let's solve the problem step-by-step:

1. Understand the problem:
- A person is standing 36 feet away from a telephone pole.
- The angle of elevation from the ground to the top of the pole is \( 30^\circ \).

2. Visualize the scenario:
- You can imagine a right triangle where:
- The horizontal leg (adjacent side) is the distance from the person to the base of the pole, which is 36 feet.
- The vertical leg (opposite side) is the height of the pole, which we need to find.
- The angle of elevation is between the ground (adjacent side) and the line of sight to the top of the pole.

3. Use trigonometry:
- For a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side.
- Mathematically, \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

4. Apply the values to the tangent function:
- Here, \( \theta = 30^\circ \) and the adjacent side is 36 feet.

[tex]\[ \tan(30^\circ) = \frac{\text{height}}{36 \text{ ft}} \][/tex]

5. Solve for the height:
- Using the known value of \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):

[tex]\[ \frac{1}{\sqrt{3}} = \frac{\text{height}}{36} \][/tex]

[tex]\[ \text{height} = 36 \cdot \frac{1}{\sqrt{3}} = 36 \cdot \frac{\sqrt{3}}{3} = 12\sqrt{3} \text{ ft} \][/tex]

Therefore, the height of the pole is \(12\sqrt{3} \) feet.

Among the given options, the correct answer is:

[tex]\[ 12 \sqrt{3} \text{ ft} \][/tex]