6loptimization) If the perimeter of the circukar Sector is fixed at lo0', what values of r and s will give the sechor the greatesst area ? Area =Ź or and S=r@, %3D where o is in radians. J dentity the primary funchion end seLondary function.
6loptimization) If the perimeter of the circukar Sector is fixed at lo0', what values of r and s will give the sechor the greatesst area ? Area =Ź or and S=r@, %3D where o is in radians. J dentity the primary funchion end seLondary function.
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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![### Optimization Problem
#### Problem Statement
**Given:** If the perimeter of the circular sector is fixed at 100 units, determine the values of \( r \) (radius) and \( s \) (arc length) that will give the sector the greatest area.
#### Formulas Provided:
- **Area of Sector:** \[ \text{Area} = \frac{1}{2} \theta r^2 \]
- **Arc Length:** \[ s = r \theta \]
where \( \theta \) is in radians.
#### Diagram Explanation:
The diagram below illustrates the circular sector with the following components:
- **Radius (r):** The distance from the center of the circle to any point on its circumference, represented in the diagram as the two radii extending from the center.
- **Arc Length (s):** The length of the curved boundary of the sector, marked in red in the diagram.
- **Angle (\(\theta\)):** The angle subtended by the arc at the center, shown in the diagram as the central angle between the two radii.
The key task is to identify and work with the primary and secondary functions to maximize the given area constraint.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e9d947d-304f-415e-95bf-140f5d3b17e4%2F3d33ad16-d5cd-4f5f-86f3-34d7f2ef727d%2F9bpy6h7_processed.png&w=3840&q=75)
Transcribed Image Text:### Optimization Problem
#### Problem Statement
**Given:** If the perimeter of the circular sector is fixed at 100 units, determine the values of \( r \) (radius) and \( s \) (arc length) that will give the sector the greatest area.
#### Formulas Provided:
- **Area of Sector:** \[ \text{Area} = \frac{1}{2} \theta r^2 \]
- **Arc Length:** \[ s = r \theta \]
where \( \theta \) is in radians.
#### Diagram Explanation:
The diagram below illustrates the circular sector with the following components:
- **Radius (r):** The distance from the center of the circle to any point on its circumference, represented in the diagram as the two radii extending from the center.
- **Arc Length (s):** The length of the curved boundary of the sector, marked in red in the diagram.
- **Angle (\(\theta\)):** The angle subtended by the arc at the center, shown in the diagram as the central angle between the two radii.
The key task is to identify and work with the primary and secondary functions to maximize the given area constraint.
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