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Suppose you are given a square with an inscribed circle as shown below. If the area of the square is 144 m² what is the area of the circle? (Use 3.14159 for .)

Suppose you are given a square with an inscribed circle as shown below. If the area of the square is 144 m² what is the area of the circle? (Use 3.14159 for .)

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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### Problem Statement Suppose you are given a square with an inscribed circle as shown below. If the area of the square is 144 m², what is the area of the circle? (Use 3.14159 for π.) ![Diagram](#) ### Diagram A square with a circle inscribed in it. The circle touches all four sides of the square. #### Given: - Area of the square = 144 m² ### Solution First, we need to find the side length of the square. The area \( A \) of a square is given by: \[ A = \text{side}^2 \] So, the side length of the square is: \[ \text{side} = \sqrt{A} = \sqrt{144} = 12 \, \text{m} \] Since the circle is inscribed in the square, the diameter of the circle is equal to the side length of the square. Therefore, the diameter \( D \) of the circle is: \[ D = 12 \, \text{m} \] The radius \( r \) of the circle is half of the diameter: \[ r = \frac{D}{2} = \frac{12}{2} = 6 \, \text{m} \] The area \( A_{\text{circle}} \) of the circle is given by: \[ A_{\text{circle}} = \pi r^2 \] Using 3.14159 for \( \pi \): \[ A_{\text{circle}} = 3.14159 \times (6)^2 = 3.14159 \times 36 = 113.10 \, \text{m}^2 \] ### Answer: \[ \text{The area of the circle is } 113.10 \, \text{m}^2 \, (\text{Write your answer to two decimal places.}) \] --- #### Notes - Time Remaining: 00:59:06 - Buttons: "Next" This problem can be utilized in the context of understanding geometric shapes and their properties, specifically focusing on the relationships between squares and circles inscribed in them.
Transcribed Image Text:### Problem Statement Suppose you are given a square with an inscribed circle as shown below. If the area of the square is 144 m², what is the area of the circle? (Use 3.14159 for π.) ![Diagram](#) ### Diagram A square with a circle inscribed in it. The circle touches all four sides of the square. #### Given: - Area of the square = 144 m² ### Solution First, we need to find the side length of the square. The area \( A \) of a square is given by: \[ A = \text{side}^2 \] So, the side length of the square is: \[ \text{side} = \sqrt{A} = \sqrt{144} = 12 \, \text{m} \] Since the circle is inscribed in the square, the diameter of the circle is equal to the side length of the square. Therefore, the diameter \( D \) of the circle is: \[ D = 12 \, \text{m} \] The radius \( r \) of the circle is half of the diameter: \[ r = \frac{D}{2} = \frac{12}{2} = 6 \, \text{m} \] The area \( A_{\text{circle}} \) of the circle is given by: \[ A_{\text{circle}} = \pi r^2 \] Using 3.14159 for \( \pi \): \[ A_{\text{circle}} = 3.14159 \times (6)^2 = 3.14159 \times 36 = 113.10 \, \text{m}^2 \] ### Answer: \[ \text{The area of the circle is } 113.10 \, \text{m}^2 \, (\text{Write your answer to two decimal places.}) \] --- #### Notes - Time Remaining: 00:59:06 - Buttons: "Next" This problem can be utilized in the context of understanding geometric shapes and their properties, specifically focusing on the relationships between squares and circles inscribed in them.
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