Owen was constructing a uniquely shaped bird feeder for his Mom for Mother's Day. The bird feeder will have a cylindrical shaped bottom with a height of 16 inches and a diameter of 7 inches. On top of the cylinder will be a hemispherical shaped dome top. Find the total amount of bird food, in cubic inches, that Owen will need to completely fill his feeder. Use T in your volume calculations and round your final answer to the nearest whole number of cubic inches. Enter only the numerical value.
Owen was constructing a uniquely shaped bird feeder for his Mom for Mother's Day. The bird feeder will have a cylindrical shaped bottom with a height of 16 inches and a diameter of 7 inches. On top of the cylinder will be a hemispherical shaped dome top. Find the total amount of bird food, in cubic inches, that Owen will need to completely fill his feeder. Use T in your volume calculations and round your final answer to the nearest whole number of cubic inches. Enter only the numerical value.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement**
Owen was constructing a uniquely shaped bird feeder for his Mom for Mother's Day. The bird feeder will have a cylindrical shaped bottom with a height of 16 inches and a diameter of 7 inches. On top of the cylinder will be a hemispherical shaped dome top. Find the total amount of bird food, in cubic inches, that Owen will need to completely fill his feeder. Use π in your volume calculations and round your final answer to the nearest whole number of cubic inches. Enter only the numerical value.
**Explanation of Calculations**
To solve this problem, you need to calculate the volume of both the cylindrical part and the hemispherical dome, then add them together to find the total volume:
1. **Volume of the Cylinder**
The volume \(V\) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
- Diameter of the cylinder = 7 inches
- Radius \(r\) = Diameter / 2 = 7 / 2 = 3.5 inches
- Height \(h\) = 16 inches
Substitute \(r\) and \(h\) into the formula:
\[ V_{cylinder} = \pi (3.5)^2 (16) \]
\[ V_{cylinder} = \pi (12.25) (16) \]
\[ V_{cylinder} = \pi (196) \]
\[ V_{cylinder} \approx 3.14159 \times 196 \]
\[ V_{cylinder} \approx 615.75 \, \text{cubic inches} \]
2. **Volume of the Hemisphere**
The volume \(V\) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Since we have a hemisphere (half of a sphere), we divide the volume by 2:
\[ V_{hemisphere} = \frac{1}{2} \left(\frac{4}{3} \pi r^3\right) \]
- Radius \(r\) = 3.5 inches (same as the cylinder)
Substitute \(r\) into the formula:
\[ V_{hemisphere} = \frac{1}{2} \left(\frac{4}{3} \pi (3.5)^3\right) \]
\[ V_{hemisphere} =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31744b2c-f7da-41b9-99c8-2f43aed6357e%2F1928c7fc-83e0-463b-8103-244c3bd16f17%2Fhanh3tn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Owen was constructing a uniquely shaped bird feeder for his Mom for Mother's Day. The bird feeder will have a cylindrical shaped bottom with a height of 16 inches and a diameter of 7 inches. On top of the cylinder will be a hemispherical shaped dome top. Find the total amount of bird food, in cubic inches, that Owen will need to completely fill his feeder. Use π in your volume calculations and round your final answer to the nearest whole number of cubic inches. Enter only the numerical value.
**Explanation of Calculations**
To solve this problem, you need to calculate the volume of both the cylindrical part and the hemispherical dome, then add them together to find the total volume:
1. **Volume of the Cylinder**
The volume \(V\) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
- Diameter of the cylinder = 7 inches
- Radius \(r\) = Diameter / 2 = 7 / 2 = 3.5 inches
- Height \(h\) = 16 inches
Substitute \(r\) and \(h\) into the formula:
\[ V_{cylinder} = \pi (3.5)^2 (16) \]
\[ V_{cylinder} = \pi (12.25) (16) \]
\[ V_{cylinder} = \pi (196) \]
\[ V_{cylinder} \approx 3.14159 \times 196 \]
\[ V_{cylinder} \approx 615.75 \, \text{cubic inches} \]
2. **Volume of the Hemisphere**
The volume \(V\) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Since we have a hemisphere (half of a sphere), we divide the volume by 2:
\[ V_{hemisphere} = \frac{1}{2} \left(\frac{4}{3} \pi r^3\right) \]
- Radius \(r\) = 3.5 inches (same as the cylinder)
Substitute \(r\) into the formula:
\[ V_{hemisphere} = \frac{1}{2} \left(\frac{4}{3} \pi (3.5)^3\right) \]
\[ V_{hemisphere} =
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