Suppose that a movie theater has a screen that is 29 feet tall. When you sit down, the bottom of the screen is 8 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure at right, 0 is the viewing angle. Suppose that you sit x feet from the screen. The viewing angle 0 is given by the function below the figure. Complete parts (a) through (c). 8 0(x) = tan x feet -1 tan 29 feet 18 feet
Suppose that a movie theater has a screen that is 29 feet tall. When you sit down, the bottom of the screen is 8 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure at right, 0 is the viewing angle. Suppose that you sit x feet from the screen. The viewing angle 0 is given by the function below the figure. Complete parts (a) through (c). 8 0(x) = tan x feet -1 tan 29 feet 18 feet
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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a. What is your viewing angle if you sit 10 feet from the screen? 15 feet? 20 feet?
B. If there are 5 feet between the screen and first row of seats and there are 3 feet between each row and row behind it, which row results in largest viewing area?
C. What value of x results in the largest viewing area?
![### Viewing Angle in a Movie Theater
#### Problem Statement
Suppose that a movie theater has a screen that is 29 feet tall. When you sit down, the bottom of the screen is 8 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle.
In the provided diagram, \( \theta \) is the viewing angle. Suppose that you sit \( x \) feet from the screen. The viewing angle \( \theta \) is given by the following function:
\[ \theta(x) = \tan^{-1} \left( \frac{37}{x} \right) - \tan^{-1} \left( \frac{8}{x} \right) \]
Complete parts (a) through (c).
#### Explanation of the Diagram
The diagram shows a right-angled triangle where:
- The height from the viewer's eye level to the bottom of the screen is 8 feet.
- The total height of the screen above the viewer's eye level is 29 feet.
- The distance from the viewer to the screen is \( x \) feet.
The diagram highlights the angles formed to calculate the viewing angle \( \theta \).
Here's a detailed breakdown of the elements in the diagram:
1. **Vertical Distance Components:**
- Bottom of the screen to eye level: 8 feet.
- Top of the screen to eye level: 29 feet.
2. **Horizontal Distance Component:**
- Distance from the viewer to the screen: \( x \) feet.
3. **Angles:**
- Viewing angle \( \theta \).
- Two angles are calculated using the tangent inverse function: one for the top of the screen and one for the bottom.
#### Function Explanation
The function \( \theta(x) \) to calculate the viewing angle is broken down into two parts:
- \( \tan^{-1} \left( \frac{37}{x} \right) \): This calculates the angle formed from your eyes to the top of the screen.
- \( \tan^{-1} \left( \frac{8}{x} \right) \): This calculates the angle formed from your eyes to the bottom of the screen.
The viewing angle is then the difference between these two angles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc944e8dc-d275-4b70-8def-6573a43e5eec%2F1b871b90-8ad7-41c8-9ece-195ef675cdaa%2F0w9xml_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Viewing Angle in a Movie Theater
#### Problem Statement
Suppose that a movie theater has a screen that is 29 feet tall. When you sit down, the bottom of the screen is 8 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle.
In the provided diagram, \( \theta \) is the viewing angle. Suppose that you sit \( x \) feet from the screen. The viewing angle \( \theta \) is given by the following function:
\[ \theta(x) = \tan^{-1} \left( \frac{37}{x} \right) - \tan^{-1} \left( \frac{8}{x} \right) \]
Complete parts (a) through (c).
#### Explanation of the Diagram
The diagram shows a right-angled triangle where:
- The height from the viewer's eye level to the bottom of the screen is 8 feet.
- The total height of the screen above the viewer's eye level is 29 feet.
- The distance from the viewer to the screen is \( x \) feet.
The diagram highlights the angles formed to calculate the viewing angle \( \theta \).
Here's a detailed breakdown of the elements in the diagram:
1. **Vertical Distance Components:**
- Bottom of the screen to eye level: 8 feet.
- Top of the screen to eye level: 29 feet.
2. **Horizontal Distance Component:**
- Distance from the viewer to the screen: \( x \) feet.
3. **Angles:**
- Viewing angle \( \theta \).
- Two angles are calculated using the tangent inverse function: one for the top of the screen and one for the bottom.
#### Function Explanation
The function \( \theta(x) \) to calculate the viewing angle is broken down into two parts:
- \( \tan^{-1} \left( \frac{37}{x} \right) \): This calculates the angle formed from your eyes to the top of the screen.
- \( \tan^{-1} \left( \frac{8}{x} \right) \): This calculates the angle formed from your eyes to the bottom of the screen.
The viewing angle is then the difference between these two angles.
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