Answer :
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]
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]
