88888 3. A person standing on level ground is 2,000 feet away from the foot of a 420-foot-tall building, as shown in the accompanying diagram. ( 20 points ) a) To the nearest degree, what is the value of x? [10 points] ot b) What is the length the wire, to the nearest foot, that is tied to the building and being held by the person on the ground?[10 points]
88888 3. A person standing on level ground is 2,000 feet away from the foot of a 420-foot-tall building, as shown in the accompanying diagram. ( 20 points ) a) To the nearest degree, what is the value of x? [10 points] ot b) What is the length the wire, to the nearest foot, that is tied to the building and being held by the person on the ground?[10 points]
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![### Educational Content: Trigonometry Problem with a Right-Angle Triangle
**Problem Statement:**
A person standing on level ground is 2,000 feet away from the foot of a 420-foot-tall building, as shown in the accompanying diagram.
**Diagram Description:**
The diagram is a right-angle triangle where:
- The height of the building is the vertical side (420 feet).
- The distance from the person to the building base is the horizontal side (2,000 feet).
- The hypotenuse is the line connecting the top of the building to the person, forming an angle \( x \) with the horizontal.
**Tasks:**
#### a) Calculate Angle \( x \)
To the nearest degree, what is the value of \( x \)?
**Given Data:**
- Opposite side (height of the building) = 420 feet
- Adjacent side (distance from the person to the building) = 2,000 feet
**Solution:**
\[ \tan(x) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{420}{2000} \]
\[ x = \tan^{-1}\left(\frac{420}{2000}\right) \]
Using a calculator to find the arctan value, we get:
\[ x \approx 12^\circ \]
#### b) Calculate the Length of the Wire
What is the length of the wire, to the nearest foot, that is tied to the building and being held by the person on the ground?
**Solution:**
We need to find the hypotenuse of the right-angle triangle.
\[
\text{Hypotenuse} = \sqrt{(\text{Opposite}^2 + \text{Adjacent}^2)} = \sqrt{(420^2 + 2000^2)}
\]
Calculating the square values and sum:
\[
420^2 = 176400 \\
2000^2 = 4000000
\]
\[
\text{Sum} = 176400 + 4000000 = 4176400
\]
\[
\sqrt{4176400} \approx 2043 \text{ feet}
\]
**Final Answers:**
a) The angle \( x \) is approximately \( 12^\circ \) (to the nearest degree).
b) The length of the wire is approximately 2,043 feet (to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b850726-b101-4cdd-96f3-5d5383803507%2Fb3ca2f8d-5b63-43d9-b21a-99dddf82d573%2F73hiuo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content: Trigonometry Problem with a Right-Angle Triangle
**Problem Statement:**
A person standing on level ground is 2,000 feet away from the foot of a 420-foot-tall building, as shown in the accompanying diagram.
**Diagram Description:**
The diagram is a right-angle triangle where:
- The height of the building is the vertical side (420 feet).
- The distance from the person to the building base is the horizontal side (2,000 feet).
- The hypotenuse is the line connecting the top of the building to the person, forming an angle \( x \) with the horizontal.
**Tasks:**
#### a) Calculate Angle \( x \)
To the nearest degree, what is the value of \( x \)?
**Given Data:**
- Opposite side (height of the building) = 420 feet
- Adjacent side (distance from the person to the building) = 2,000 feet
**Solution:**
\[ \tan(x) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{420}{2000} \]
\[ x = \tan^{-1}\left(\frac{420}{2000}\right) \]
Using a calculator to find the arctan value, we get:
\[ x \approx 12^\circ \]
#### b) Calculate the Length of the Wire
What is the length of the wire, to the nearest foot, that is tied to the building and being held by the person on the ground?
**Solution:**
We need to find the hypotenuse of the right-angle triangle.
\[
\text{Hypotenuse} = \sqrt{(\text{Opposite}^2 + \text{Adjacent}^2)} = \sqrt{(420^2 + 2000^2)}
\]
Calculating the square values and sum:
\[
420^2 = 176400 \\
2000^2 = 4000000
\]
\[
\text{Sum} = 176400 + 4000000 = 4176400
\]
\[
\sqrt{4176400} \approx 2043 \text{ feet}
\]
**Final Answers:**
a) The angle \( x \) is approximately \( 12^\circ \) (to the nearest degree).
b) The length of the wire is approximately 2,043 feet (to
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