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12. What is the area of the shaded area below? Answer in terms of it. Area = 8 cm 3 em

12. What is the area of the shaded area below? Answer in terms of it. Area = 8 cm 3 em

Elementary Geometry For College Students, 7e
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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**Mathematics Problem: Calculating the Area of a Shaded Region** **Problem 12: What is the area of the shaded area below? Answer in terms of π.** **Given Information:** - Diagram Description: There are two concentric circles (one circle within another). - The radius of the larger circle: 8 cm - The radius of the smaller circle: 3 cm **Step-by-Step Solution:** To find the area of the shaded region, we need to subtract the area of the smaller circle from the area of the larger circle. 1. **Calculate the area of the larger circle:** The area \(A\) of a circle is given by the formula: \[ A = \pi r^2 \] where \(r\) is the radius. For the larger circle: \[ \text{Radius} = 8 \text{ cm} \] \[ A_{\text{large}} = \pi (8)^2 = 64\pi \text{ cm}^2 \] 2. **Calculate the area of the smaller circle:** For the smaller circle: \[ \text{Radius} = 3 \text{ cm} \] \[ A_{\text{small}} = \pi (3)^2 = 9\pi \text{ cm}^2 \] 3. **Determine the area of the shaded region:** \[ A_{\text{shaded}} = A_{\text{large}} - A_{\text{small}} \] \[ A_{\text{shaded}} = 64\pi - 9\pi = 55\pi \text{ cm}^2 \] Thus, the area of the shaded region is \(55\pi\) square centimeters. **Final Answer:** Area = \(55\pi \text{ cm}^2\).
Transcribed Image Text:**Mathematics Problem: Calculating the Area of a Shaded Region** **Problem 12: What is the area of the shaded area below? Answer in terms of π.** **Given Information:** - Diagram Description: There are two concentric circles (one circle within another). - The radius of the larger circle: 8 cm - The radius of the smaller circle: 3 cm **Step-by-Step Solution:** To find the area of the shaded region, we need to subtract the area of the smaller circle from the area of the larger circle. 1. **Calculate the area of the larger circle:** The area \(A\) of a circle is given by the formula: \[ A = \pi r^2 \] where \(r\) is the radius. For the larger circle: \[ \text{Radius} = 8 \text{ cm} \] \[ A_{\text{large}} = \pi (8)^2 = 64\pi \text{ cm}^2 \] 2. **Calculate the area of the smaller circle:** For the smaller circle: \[ \text{Radius} = 3 \text{ cm} \] \[ A_{\text{small}} = \pi (3)^2 = 9\pi \text{ cm}^2 \] 3. **Determine the area of the shaded region:** \[ A_{\text{shaded}} = A_{\text{large}} - A_{\text{small}} \] \[ A_{\text{shaded}} = 64\pi - 9\pi = 55\pi \text{ cm}^2 \] Thus, the area of the shaded region is \(55\pi\) square centimeters. **Final Answer:** Area = \(55\pi \text{ cm}^2\).
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