- A wedge has been cut from a cylindrical container that has a radius of 6 inches and a height of 15 inches. Three-fourths of a cylinder remains. Find the surface area of the remaining part of the cylinder. Round to the nearest 10th. in2 6 in. 15 in.
- A wedge has been cut from a cylindrical container that has a radius of 6 inches and a height of 15 inches. Three-fourths of a cylinder remains. Find the surface area of the remaining part of the cylinder. Round to the nearest 10th. in2 6 in. 15 in.
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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![**Problem Statement:**
4. A wedge has been cut from a cylindrical container that has a radius of 6 inches and a height of 15 inches. Three-fourths of a cylinder remains. Find the surface area of the remaining part of the cylinder. Round to the nearest 10th.
**Diagram Explanation:**
A cylindrical container is shown with a wedge removed, indicating that three-fourths of the cylinder remains. The given measurements are:
- Radius (r) = 6 inches
- Height (h) = 15 inches
**Solution Steps:**
Overall, this problem involves calculating the surface area of a portion of a cylinder after a wedge is removed. The key steps involve:
1. Calculate the complete surface area of the original cylinder.
2. Determine the proportion of the surface area that remains.
3. Adjust and find the surface area of the remaining three-fourths of the cylinder.
**Useful Formulas:**
1. Surface area of a cylinder (without the top and bottom) = \( 2\pi rh \)
2. Surface area of the top and bottom of the cylinder = \( 2\pi r^2 \)
**Step-by-step Calculation:**
1. Surface area of the cylindrical side:
\[ 2\pi rh = 2\pi \times 6 \times 15 = 180\pi \ \text{square inches} \]
2. Surface area of the top and bottom:
\[ 2\pi r^2 = 2\pi \times 6^2 = 72\pi \ \text{square inches} \]
3. Total surface area of the whole cylinder:
\[ \text{Total} = 180\pi + 72\pi = 252\pi \ \text{square inches} \]
4. Since only three-fourths of the cylinder remains, we take three-fourths of the total surface area:
\[ \text{Remaining surface area} = \frac{3}{4} \times 252\pi = 189\pi \ \text{square inches} \]
5. Convert the final surface area to a decimal value and round to the nearest 10th:
\[ 189\pi \approx 593.6 \ \text{square inches} \]
**Answer:**
The surface area of the remaining part of the cylinder is approximately \( 593.6 \ \text](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31516e53-ca95-4b50-978f-e7886c753a90%2F259be48a-10ac-4c6e-9f44-cbc11b0e4607%2Fvodjsgm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
4. A wedge has been cut from a cylindrical container that has a radius of 6 inches and a height of 15 inches. Three-fourths of a cylinder remains. Find the surface area of the remaining part of the cylinder. Round to the nearest 10th.
**Diagram Explanation:**
A cylindrical container is shown with a wedge removed, indicating that three-fourths of the cylinder remains. The given measurements are:
- Radius (r) = 6 inches
- Height (h) = 15 inches
**Solution Steps:**
Overall, this problem involves calculating the surface area of a portion of a cylinder after a wedge is removed. The key steps involve:
1. Calculate the complete surface area of the original cylinder.
2. Determine the proportion of the surface area that remains.
3. Adjust and find the surface area of the remaining three-fourths of the cylinder.
**Useful Formulas:**
1. Surface area of a cylinder (without the top and bottom) = \( 2\pi rh \)
2. Surface area of the top and bottom of the cylinder = \( 2\pi r^2 \)
**Step-by-step Calculation:**
1. Surface area of the cylindrical side:
\[ 2\pi rh = 2\pi \times 6 \times 15 = 180\pi \ \text{square inches} \]
2. Surface area of the top and bottom:
\[ 2\pi r^2 = 2\pi \times 6^2 = 72\pi \ \text{square inches} \]
3. Total surface area of the whole cylinder:
\[ \text{Total} = 180\pi + 72\pi = 252\pi \ \text{square inches} \]
4. Since only three-fourths of the cylinder remains, we take three-fourths of the total surface area:
\[ \text{Remaining surface area} = \frac{3}{4} \times 252\pi = 189\pi \ \text{square inches} \]
5. Convert the final surface area to a decimal value and round to the nearest 10th:
\[ 189\pi \approx 593.6 \ \text{square inches} \]
**Answer:**
The surface area of the remaining part of the cylinder is approximately \( 593.6 \ \text
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