Chapter5: Exponential And Logarithmic Functions
Section: Chapter Questions
Problem 22PS: In a classroom designed for 30 students, the air conditioning system can move 450 cubic feet of air...
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![**Finding the Volume of a Composite Shape**
The image depicts a composite shape made up of a cylinder with a hemisphere on top. The cylinder has a height of 24 feet and a diameter of 16 feet.
To calculate the volume of this composite shape, follow these steps:
1. **Volume of the Cylinder:**
- The radius (r) of the cylinder is half of the diameter, so \( r = \frac{16}{2} = 8 \) feet.
- The height (h) of the cylinder is 24 feet.
- The formula for the volume \( V \) of a cylinder is:
\[
V = \pi r^2 h
\]
- Plug in the values:
\[
V_\text{cylinder} = \pi \times 8^2 \times 24 = \pi \times 64 \times 24 = 1536\pi \approx 4824.84 \text{ cubic feet}
\]
2. **Volume of the Hemisphere:**
- The radius (r) of the hemisphere is the same as the cylinder, \( r = 8 \) feet.
- The formula for the volume \( V \) of a sphere is:
\[
V = \frac{4}{3} \pi r^3
\]
- Since we only have a hemisphere (half of a sphere), the volume is:
\[
V_\text{hemisphere} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]
- Plug in the values:
\[
V_\text{hemisphere} = \frac{2}{3} \pi \times 8^3 = \frac{2}{3} \pi \times 512 = \frac{1024}{3} \pi \approx 1073.38 \text{ cubic feet}
\]
3. **Total Volume:**
- Sum the volumes of the cylinder and the hemisphere:
\[
V_\text{total} = V_\text{cylinder} + V_\text{hemisphere} = 1536\pi + \frac{1024}{3}\pi \approx 4824.84 + 1073](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb73d9f23-fb52-4407-b592-aeda313b8dae%2F2b3aa3de-2c46-4c0c-b1ca-4d5167f653e9%2Fru5cfu7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Volume of a Composite Shape**
The image depicts a composite shape made up of a cylinder with a hemisphere on top. The cylinder has a height of 24 feet and a diameter of 16 feet.
To calculate the volume of this composite shape, follow these steps:
1. **Volume of the Cylinder:**
- The radius (r) of the cylinder is half of the diameter, so \( r = \frac{16}{2} = 8 \) feet.
- The height (h) of the cylinder is 24 feet.
- The formula for the volume \( V \) of a cylinder is:
\[
V = \pi r^2 h
\]
- Plug in the values:
\[
V_\text{cylinder} = \pi \times 8^2 \times 24 = \pi \times 64 \times 24 = 1536\pi \approx 4824.84 \text{ cubic feet}
\]
2. **Volume of the Hemisphere:**
- The radius (r) of the hemisphere is the same as the cylinder, \( r = 8 \) feet.
- The formula for the volume \( V \) of a sphere is:
\[
V = \frac{4}{3} \pi r^3
\]
- Since we only have a hemisphere (half of a sphere), the volume is:
\[
V_\text{hemisphere} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]
- Plug in the values:
\[
V_\text{hemisphere} = \frac{2}{3} \pi \times 8^3 = \frac{2}{3} \pi \times 512 = \frac{1024}{3} \pi \approx 1073.38 \text{ cubic feet}
\]
3. **Total Volume:**
- Sum the volumes of the cylinder and the hemisphere:
\[
V_\text{total} = V_\text{cylinder} + V_\text{hemisphere} = 1536\pi + \frac{1024}{3}\pi \approx 4824.84 + 1073
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