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Find the surface area for the composite figure which is a cylinder with a hemispherical hole. Leave answer in terms of t. 2 in. 8 in. square inches 3 in.

Find the surface area for the composite figure which is a cylinder with a hemispherical hole. Leave answer in terms of t. 2 in. 8 in. square inches 3 in.

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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**Finding the Surface Area of a Composite Figure: Cylinder with Hemispherical Hole** **Problem Statement:** Find the surface area for the composite figure, which is a cylinder with a hemispherical hole. Leave your answer in terms of \(\pi\). **Figure Details:** 1. The cylinder has a height of 3 inches. 2. The radius of the cylinder is 8 inches. 3. The hemispherical hole has a radius of 2 inches. **Visual Diagram Description:** The provided diagram illustrates a cylinder with a hemispherical hole centered on its top surface. - The height (depth) of the cylinder is labeled as 3 inches. - The radius of the cylinder is labeled as 8 inches. - The radius of the hemispherical hole is labeled as 2 inches. - The hemispherical hole is positioned centrally on the top face of the cylinder. **Solution Approach:** To find the surface area of this composite solid, follow these steps: 1. **Surface Area of the Cylinder (without the hole):** - Lateral Surface Area of the Cylinder: \(2\pi rh\) - Bottom Surface Area of the Cylinder: \(\pi r^2\) - Top Surface Area of the Cylinder: \(\pi r^2\) Here, \(r = 8 \text{ inches}\) and \(h = 3 \text{ inches}\). 2. **Surface Area of the Hemispherical Hole:** - The surface area of the hemisphere (exterior surface): \(2\pi r^2\) Since this is a hole, we subtract this surface area from the top surface area of the cylinder. 3. **Total Surface Area:** - Total Surface Area = Lateral Surface Area of Cylinder + Bottom Surface Area of Cylinder + Top Surface Area of Cylinder - Area of the Circular Base of the Hemisphere + Surface Area of the Hemisphere **Mathematical Calculation:** 1. Lateral Surface Area of the Cylinder: \( \text{LSA} = 2\pi rh = 2\pi(8)(3) = 48\pi \) 2. Bottom Surface Area of the Cylinder \( \pi r^2 \): \( \text{BSA} = \pi(8)^2 = 64\pi \) 3. Top Surface Area of the Cylinder without the hemisphere hole: \( \text{
Transcribed Image Text:**Finding the Surface Area of a Composite Figure: Cylinder with Hemispherical Hole** **Problem Statement:** Find the surface area for the composite figure, which is a cylinder with a hemispherical hole. Leave your answer in terms of \(\pi\). **Figure Details:** 1. The cylinder has a height of 3 inches. 2. The radius of the cylinder is 8 inches. 3. The hemispherical hole has a radius of 2 inches. **Visual Diagram Description:** The provided diagram illustrates a cylinder with a hemispherical hole centered on its top surface. - The height (depth) of the cylinder is labeled as 3 inches. - The radius of the cylinder is labeled as 8 inches. - The radius of the hemispherical hole is labeled as 2 inches. - The hemispherical hole is positioned centrally on the top face of the cylinder. **Solution Approach:** To find the surface area of this composite solid, follow these steps: 1. **Surface Area of the Cylinder (without the hole):** - Lateral Surface Area of the Cylinder: \(2\pi rh\) - Bottom Surface Area of the Cylinder: \(\pi r^2\) - Top Surface Area of the Cylinder: \(\pi r^2\) Here, \(r = 8 \text{ inches}\) and \(h = 3 \text{ inches}\). 2. **Surface Area of the Hemispherical Hole:** - The surface area of the hemisphere (exterior surface): \(2\pi r^2\) Since this is a hole, we subtract this surface area from the top surface area of the cylinder. 3. **Total Surface Area:** - Total Surface Area = Lateral Surface Area of Cylinder + Bottom Surface Area of Cylinder + Top Surface Area of Cylinder - Area of the Circular Base of the Hemisphere + Surface Area of the Hemisphere **Mathematical Calculation:** 1. Lateral Surface Area of the Cylinder: \( \text{LSA} = 2\pi rh = 2\pi(8)(3) = 48\pi \) 2. Bottom Surface Area of the Cylinder \( \pi r^2 \): \( \text{BSA} = \pi(8)^2 = 64\pi \) 3. Top Surface Area of the Cylinder without the hemisphere hole: \( \text{
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