A 200-foot-high construction is anchored to the ground by a wire. The wire is fastened to the ground 25 feet from the base of the construction. What is the angle the wire forms with the ground? Round your answer to the nearest degree. 23° 45° 083° 115°
A 200-foot-high construction is anchored to the ground by a wire. The wire is fastened to the ground 25 feet from the base of the construction. What is the angle the wire forms with the ground? Round your answer to the nearest degree. 23° 45° 083° 115°
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°.
Related questions
Question
![**Problem Statement:**
A 200-foot-high construction is anchored to the ground by a wire. The wire is fastened to the ground 25 feet from the base of the construction. What is the angle the wire forms with the ground? Round your answer to the nearest degree.
**Options:**
- 23°
- 45°
- 83°
- 115°
**Answer Choices Explanation:**
You need to determine the angle of elevation formed between the ground and the wire. This involves using trigonometric ratios.
**Detailed Explanation:**
In this problem, you can use the tangent function, because you know the lengths of the opposite and adjacent sides relative to the angle formed between the wire and the ground.
Given:
- Height of the construction (opposite side) = 200 feet
- Distance from the base where the wire is fastened (adjacent side) = 25 feet
The formula for the tangent of angle θ is:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{200}{25} = 8 \]
To find the angle θ, take the arctangent (inverse tangent) of both sides:
\[ \theta = \tan^{-1}(8) \]
Using a calculator to find θ:
\[ \theta \approx 82.874° \]
Rounding to the nearest degree, you get:
\[ \theta \approx 83° \]
Therefore, the correct answer is 83°.
**Graph/Diagram Description:**
There isn't a graph or diagram provided in the image, but if you were to visualize it, you would see a right triangle where:
- One leg of the triangle represents the height of the construction (200 feet),
- The other leg represents the horizontal distance from the base to where the wire is fastened (25 feet),
- The hypotenuse of the triangle is the wire.
The angle we are solving for is the angle of elevation between the hypotenuse (the wire) and the adjacent side (the ground).
**Interactive Features:**
- The problem statement is straightforward with an accompanying list of multiple-choice options.
- Below the problem, there are standard navigation buttons such as "NEXT QUESTION," "ASK FOR HELP," and "TURN IT IN," which likely allow the user to navigate through a sequence of problems, get assistance, or submit their answer, respectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d8335b2-cc2b-436a-9010-3f7343f930df%2F89b9b290-444a-46b9-b43d-6b37061ec501%2F7lir8ta_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A 200-foot-high construction is anchored to the ground by a wire. The wire is fastened to the ground 25 feet from the base of the construction. What is the angle the wire forms with the ground? Round your answer to the nearest degree.
**Options:**
- 23°
- 45°
- 83°
- 115°
**Answer Choices Explanation:**
You need to determine the angle of elevation formed between the ground and the wire. This involves using trigonometric ratios.
**Detailed Explanation:**
In this problem, you can use the tangent function, because you know the lengths of the opposite and adjacent sides relative to the angle formed between the wire and the ground.
Given:
- Height of the construction (opposite side) = 200 feet
- Distance from the base where the wire is fastened (adjacent side) = 25 feet
The formula for the tangent of angle θ is:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{200}{25} = 8 \]
To find the angle θ, take the arctangent (inverse tangent) of both sides:
\[ \theta = \tan^{-1}(8) \]
Using a calculator to find θ:
\[ \theta \approx 82.874° \]
Rounding to the nearest degree, you get:
\[ \theta \approx 83° \]
Therefore, the correct answer is 83°.
**Graph/Diagram Description:**
There isn't a graph or diagram provided in the image, but if you were to visualize it, you would see a right triangle where:
- One leg of the triangle represents the height of the construction (200 feet),
- The other leg represents the horizontal distance from the base to where the wire is fastened (25 feet),
- The hypotenuse of the triangle is the wire.
The angle we are solving for is the angle of elevation between the hypotenuse (the wire) and the adjacent side (the ground).
**Interactive Features:**
- The problem statement is straightforward with an accompanying list of multiple-choice options.
- Below the problem, there are standard navigation buttons such as "NEXT QUESTION," "ASK FOR HELP," and "TURN IT IN," which likely allow the user to navigate through a sequence of problems, get assistance, or submit their answer, respectively.
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