To find the volume of air in the pipe, we need to determine the difference between the volume of the external cylinder (considering the pipe's outer radius) and the volume of the internal cylinder (the hollow part of the pipe considering the inner radius). Given the pipe's length is 22 meters, we must first convert this to centimeters since the radii are given in centimeters.
1. Convert the Length of the Pipe:
- The length of the pipe is [tex]\( 22 \)[/tex] meters. Since there are [tex]\( 100 \)[/tex] centimeters in a meter,
[tex]\[
\text{Length in centimeters} = 22 \times 100 = 2200 \text{ cm}
\][/tex]
2. Calculate the Volume of the External Cylinder:
- The formula for the volume of a cylinder is [tex]\( V = \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height (or length) of the cylinder.
- For the external cylinder, the radius [tex]\( r_{ext} = 2.8 \)[/tex] cm and the height [tex]\( h = 2200 \)[/tex] cm:
[tex]\[
V_{ext} = \pi \times (2.8)^2 \times 2200
\][/tex]
- This calculation gives us the volume of the external cylinder:
[tex]\[
V_{ext} \approx 54186.19 \text{ cubic centimeters}
\][/tex]
3. Calculate the Volume of the Internal Cylinder:
- For the internal cylinder, the radius [tex]\( r_{int} = 2.1 \)[/tex] cm and the height [tex]\( h = 2200 \)[/tex] cm:
[tex]\[
V_{int} = \pi \times (2.1)^2 \times 2200
\][/tex]
- This calculation gives us the volume of the internal cylinder:
[tex]\[
V_{int} \approx 30479.73 \text{ cubic centimeters}
\][/tex]
4. Calculate the Volume of Air in the Pipe:
- The volume of air is the difference between the volumes of the external and internal cylinders:
[tex]\[
V_{air} = V_{ext} - V_{int} \approx 54186.19 - 30479.73
\][/tex]
- Simplifying this:
[tex]\[
V_{air} \approx 23706.46 \text{ cubic centimeters}
\][/tex]
Therefore, the volume of air in the pipe is approximately [tex]\( 23706.46 \)[/tex] cubic centimeters.
