Solve each problem. See Examples 3 and 4. Height of an Antenna A scanner antenna is on top of the center of a house. The angle of elevation from a point 28.0 m from the center of the house to the top of the antenna is 27°10', and the angle of elevation to the bottom of the antenna is 18°10'. Find the height of the antenna.

Verified step by step guidance

1. Identify the points involved: the point on the ground 28.0 m from the house center, the bottom of the antenna (top of the house), and the top of the antenna. The horizontal distance from the observation point to the house center is 28.0 m.
2. Convert the given angles from degrees and minutes to decimal degrees or use them as is in trigonometric functions. For example, 27°10' means 27 degrees and 10 minutes, where 1 minute = 1/60 degrees.
3. Use the tangent function, which relates the angle of elevation to the opposite side (height) and adjacent side (horizontal distance). For the bottom of the antenna (top of the house), set up the equation: \(\tan(18^{\circ}10') = \frac{h_{house}}{28.0}\), where \(h_{house}\) is the height of the house.
4. Similarly, for the top of the antenna, set up the equation: \(\tan(27^{\circ}10') = \frac{h_{house} + h_{antenna}}{28.0}\), where \(h_{antenna}\) is the height of the antenna.
5. Solve the first equation for \(h_{house}\), then substitute into the second equation to solve for \(h_{house} + h_{antenna}\). Finally, subtract \(h_{house}\) from this result to find the height of the antenna: \(h_{antenna} = (h_{house} + h_{antenna}) - h_{house}\).

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed between the horizontal line from the observer's eye and the line of sight to an object above the horizontal. It helps determine the height of objects by relating distances and angles in right triangles.

Right Triangle Trigonometry

Right triangle trigonometry uses sine, cosine, and tangent ratios to relate the angles and sides of right triangles. In this problem, tangent is used to connect the height of the antenna and the horizontal distance from the observer.

Difference of Heights Using Angles

By calculating the heights corresponding to two different angles of elevation (top and bottom of the antenna) from the same horizontal distance, the height of the antenna is found by subtracting the lower height from the higher one.