Two wires, a = 120 ft apart, tether a balloon to the ground, as shown. How high is the balloon above the ground? (Round your answer to the nearest whole number.) h = ft
Two wires, a = 120 ft apart, tether a balloon to the ground, as shown. How high is the balloon above the ground? (Round your answer to the nearest whole number.) h = ft
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Two wires, a = 120 ft apart, tether a balloon to the ground, as shown. How high is the balloon above the ground? (Round your answer to the nearest whole number.) h = ft
![### Problem: Determining the height of the Balloon
In this diagram, we aim to determine the height (\( h \)) of a hot air balloon above the ground. The figure illustrates a scenario in which the height of the balloon is measured using two different angles of elevation from two distinct points on the ground.
The points on the ground are denoted such that the angle of elevation to the balloon from the first point is 75°, and from the second point, it is 85°. The horizontal distance between these two points on the ground is represented by \( a \).
To find the height \( h \) of the hot air balloon, we can use trigonometric relationships. Specifically, we will employ the tangent function which relates the angle of elevation to the height of the balloon and the distance from the point directly below the balloon to each observation point.
Given:
1. Angle of elevation from the first point: 75°
2. Angle of elevation from the second point: 85°
3. Distance between the two points on the ground: \( a \) (horizontal distance)
#### Step-by-Step Solution:
1. **Using tan(75°)**:
\[ \tan(75°) = \frac{h}{d} \]
Therefore,
\[ h = d \cdot \tan(75°) \]
2. **Using tan(85°)**:
\[ \tan(85°) = \frac{h}{d - a} \]
Therefore,
\[ h = (d - a) \cdot \tan(85°) \]
Since both equations represent \( h \), we can set them equal to each other:
\[ d \cdot \tan(75°) = (d - a) \cdot \tan(85°) \]
By solving this equation for \( d \), we can substitute back to find the height \( h \) of the balloon.
This diagram is an excellent example of how trigonometric functions can be applied to solve real-world elevation problems. By understanding the relationships between angles, distances, and heights, students can deepen their understanding of trigonometry and its applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb62fb42d-f2b4-4330-85ad-951614f13399%2Fead535da-d473-41b2-92e5-650b5b74c7ab%2Fasvli1o.gif&w=3840&q=75)
Transcribed Image Text:### Problem: Determining the height of the Balloon
In this diagram, we aim to determine the height (\( h \)) of a hot air balloon above the ground. The figure illustrates a scenario in which the height of the balloon is measured using two different angles of elevation from two distinct points on the ground.
The points on the ground are denoted such that the angle of elevation to the balloon from the first point is 75°, and from the second point, it is 85°. The horizontal distance between these two points on the ground is represented by \( a \).
To find the height \( h \) of the hot air balloon, we can use trigonometric relationships. Specifically, we will employ the tangent function which relates the angle of elevation to the height of the balloon and the distance from the point directly below the balloon to each observation point.
Given:
1. Angle of elevation from the first point: 75°
2. Angle of elevation from the second point: 85°
3. Distance between the two points on the ground: \( a \) (horizontal distance)
#### Step-by-Step Solution:
1. **Using tan(75°)**:
\[ \tan(75°) = \frac{h}{d} \]
Therefore,
\[ h = d \cdot \tan(75°) \]
2. **Using tan(85°)**:
\[ \tan(85°) = \frac{h}{d - a} \]
Therefore,
\[ h = (d - a) \cdot \tan(85°) \]
Since both equations represent \( h \), we can set them equal to each other:
\[ d \cdot \tan(75°) = (d - a) \cdot \tan(85°) \]
By solving this equation for \( d \), we can substitute back to find the height \( h \) of the balloon.
This diagram is an excellent example of how trigonometric functions can be applied to solve real-world elevation problems. By understanding the relationships between angles, distances, and heights, students can deepen their understanding of trigonometry and its applications.
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