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A triangle is placed in a semicircle with a radius of 6ft, as shown below. Find the area of the shaded region. Use the value 3.14 for π, and do not round your answer. Be sure to include the correct unit in your answer. Please see attached image for semi circle:

A triangle is placed in a semicircle with a radius of 6ft, as shown below. Find the area of the shaded region. Use the value 3.14 for π, and do not round your answer. Be sure to include the correct unit in your answer. Please see attached image for semi circle:

Calculus: Early Transcendentals
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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Hello, could you please help with this? I would like to see if I got it right, thanks!

A triangle is placed in a semicircle with a radius of 6ft, as shown below. Find the area of the shaded region.

Use the value 3.14 for π, and do not round your answer. Be sure to include the correct unit in your answer.

Please see attached image for semi circle:

Thank you for your help!

### Calculating the Radius of a Semicircle from the Height In this educational content, we will discuss how to find the radius of a semicircle given the height of the triangle inscribed within the semicircle. #### Diagram Description: The diagram provided is of a semicircle with an inscribed triangle where: - The base of the triangle coincides with the diameter of the semicircle. - The height of the triangle is given as 6 feet, which extends from the midpoint of the base (diameter) perpendicularly to the circumference of the semicircle. ### Step-by-Step Calculation: 1. **Understanding the Relationship:** - The height given in the triangle (6 feet) is the perpendicular distance from the base to the top of the semicircle, which represents the radius of the semicircle. 2. **Using the Radius in Calculation:** - In a semicircle, the height of the inscribed triangle is equal to the radius of the semicircle. - Therefore, the radius \( r \) of the semicircle can be directly taken as the height provided. 3. **Result:** - Thus, the radius \( r \) of the semicircle is \( 6 \) feet. ### Conclusion By understanding the geometrical properties of a semicircle and the inscribed triangle, we can deduce that the height of the triangle provided in the diagram directly gives us the radius of the semicircle. Therefore, the radius of the semicircle in this case is 6 feet. ### Practical Applications: This concept is fundamental in various geometrical problems and real-life applications, including engineering design, architecture, and any field requiring spatial understanding of geometric figures. Let's move on to solving more complex geometrical problems using similar principles!
Transcribed Image Text:### Calculating the Radius of a Semicircle from the Height In this educational content, we will discuss how to find the radius of a semicircle given the height of the triangle inscribed within the semicircle. #### Diagram Description: The diagram provided is of a semicircle with an inscribed triangle where: - The base of the triangle coincides with the diameter of the semicircle. - The height of the triangle is given as 6 feet, which extends from the midpoint of the base (diameter) perpendicularly to the circumference of the semicircle. ### Step-by-Step Calculation: 1. **Understanding the Relationship:** - The height given in the triangle (6 feet) is the perpendicular distance from the base to the top of the semicircle, which represents the radius of the semicircle. 2. **Using the Radius in Calculation:** - In a semicircle, the height of the inscribed triangle is equal to the radius of the semicircle. - Therefore, the radius \( r \) of the semicircle can be directly taken as the height provided. 3. **Result:** - Thus, the radius \( r \) of the semicircle is \( 6 \) feet. ### Conclusion By understanding the geometrical properties of a semicircle and the inscribed triangle, we can deduce that the height of the triangle provided in the diagram directly gives us the radius of the semicircle. Therefore, the radius of the semicircle in this case is 6 feet. ### Practical Applications: This concept is fundamental in various geometrical problems and real-life applications, including engineering design, architecture, and any field requiring spatial understanding of geometric figures. Let's move on to solving more complex geometrical problems using similar principles!
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