To determine the volume of metal needed to make the cylindrical metal pipe, we can follow these steps:
1. Calculate the Volume of the Outer 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.
- The diameter of the outer cylinder is 20 millimeters, so the radius [tex]\( r_{\text{outer}} \)[/tex] is half of that:
[tex]\[
r_{\text{outer}} = \frac{20}{2} = 10 \text{ millimeters}
\][/tex]
- The height [tex]\( h \)[/tex] of the cylinder is 21 millimeters.
Plugging these values into the formula for volume:
[tex]\[
V_{\text{outer}} = \pi (10)^2 (21)
= 2100 \pi \text{ cubic millimeters}
\][/tex]
2. Calculate the Volume of the Inner Hole:
- The radius [tex]\( r_{\text{inner}} \)[/tex] of the inner hole is 6 millimeters.
- The height [tex]\( h \)[/tex] remains 21 millimeters.
Using the volume formula:
[tex]\[
V_{\text{inner}} = \pi (6)^2 (21)
= 21 \pi (36)
= 756 \pi \text{ cubic millimeters}
\][/tex]
3. Calculate the Volume of the Metal Needed:
The volume of metal needed is the volume of the outer cylinder minus the volume of the inner hole:
[tex]\[
V_{\text{metal}} = V_{\text{outer}} - V_{\text{inner}}
= 2100 \pi - 756 \pi
= (2100 - 756) \pi
= 1344 \pi \text{ cubic millimeters}
\][/tex]
Based on the explanation above, the volume of metal needed to make the pipe is represented by the following expressions:
- [tex]\( 21 \pi (10)^2 - 21 \pi (6)^2 \)[/tex]
- [tex]\( 2100 \pi - 756 \pi \)[/tex]
These expressions correctly represent the calculations performed to find the volume of metal.
The correct options are:
1. [tex]\( 21 \pi (10)^2 - 21 \pi (6)^2 \)[/tex]
2. [tex]\( 2100 \pi - 756 \pi \)[/tex]
