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Chapter 6: Problem 113

From a point on level ground 120 feet from the base of a tower, the angle of elevation is \(48.3^{\circ} .\) Approximate the height of the tower to the nearest foot. (Section 4.8 Example 2 )

Short Answer

Expert verified
The height of the tower is approximately 116 feet.

Step by step solution

01

Recognize the problem type

Identify that this is a problem dealing with right triangle trigonometry. An angle of elevation, a vertical height (which we're trying to find), and a horizontal distance together form a right triangle.
02

Use the tangent function

In a right triangle, the tangent of an angle is the ratio of the length of the side opposite the angle to the length of the adjacent side. We're given the angle of elevation \(48.3^{\circ}\) and the distance from the tower, which is the adjacent side. The height of the tower is the opposite side. So, set up the equation \(\tan(48.3^{\circ}) = \frac{unknown \ height}{120}\).
03

Solve for the unknown height

Get the height of the tower alone by multiplying both sides of the equation by 120, giving the equation for the unknown height as: \(\text{height} = 120 \times \tan(48.3) \). Calculate the right side of this equation using a calculator.
04

Round the result

Approximate the result to the nearest foot as asked in the question.

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