A person is standing exactly 36 ft 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. 12 ft
B. [tex]$12 \sqrt{3} ft$[/tex]
C. 18 ft
D. [tex]$18 \sqrt{2} ft$[/tex]
Answer (2)
The height of the telephone pole is calculated using the tangent function, which gives us the result of 12 3 ft. Therefore, the correct answer is option B.
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Recognize the problem as a right triangle scenario where the height of the pole is opposite to the given angle.
Apply the tangent function: tan ( 3 0 ∘ ) = 36 h .
Solve for h : h = 36 tan ( 3 0 ∘ ) = 36 ⋅ 3 3 .
Simplify to find the height: h = 12 3 . The height of the pole is 12 3 f t .
Explanation
Problem Analysis Let's analyze the problem. We have a right triangle formed by the telephone pole, the ground, and the line of sight from the person to the top of the pole. The distance from the person to the pole is the base of the triangle, which is 36 ft. The angle of elevation is 3 0 ∘ . We need to find the height of the pole, which is the opposite side of the triangle.
Applying Tangent Function We can use the tangent function to relate the angle of elevation, the distance from the pole, and the height of the pole. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case: tan ( θ ) = adjacent opposite tan ( 3 0 ∘ ) = 36 h where h is the height of the pole.
Solving for Height Now, we need to solve for h . Multiply both sides of the equation by 36: h = 36 tan ( 3 0 ∘ ) We know that tan ( 3 0 ∘ ) = 3 3 . Substitute this value into the equation: h = 36 ⋅ 3 3 h = 12 3 So, the height of the pole is 12 3 ft.
Final Answer Therefore, the height of the telephone pole is 12 3 feet.
Examples
Imagine you're an architect designing a building and need to determine the height of a flagpole. You stand a certain distance away from the flagpole and measure the angle of elevation to the top. Using trigonometry, specifically the tangent function, you can calculate the height of the flagpole. This principle applies to various real-world scenarios, such as determining the height of trees, buildings, or mountains.
