Let's solve the problem step-by-step.
1. Understanding the Given Dimensions:
- The outer cylinder has a diameter of 20 millimeters and a height of 21 millimeters.
- The hole inside the cylinder has a radius of 6 millimeters.
2. Calculate the Radius of the Outer Cylinder:
- The radius of the outer cylinder is half of the diameter.
[tex]\[
\text{Radius of the outer cylinder} = \frac{20}{2} = 10 \text{ millimeters}
\][/tex]
3. Volume of the Outer Cylinder:
- The volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[
V = \pi r^2 h
\][/tex]
- Applying this formula to the outer cylinder:
[tex]\[
\text{Volume of the outer cylinder} = \pi (10)^2 (21)
\][/tex]
[tex]\[
= 2100 \pi \text{ cubic millimeters}
\][/tex]
4. Volume of the Cylindrical Hole:
- The volume of the hole, which is also cylindrical, can be computed using the same formula:
[tex]\[
V = \pi r^2 h
\][/tex]
- Applying this formula to the hole:
[tex]\[
\text{Volume of the hole} = \pi (6)^2 (21)
\][/tex]
[tex]\[
= 756 \pi \text{ cubic millimeters}
\][/tex]
5. Volume of Metal Needed:
- The volume of metal needed to make the pipe is the volume of the outer cylinder minus the volume of the hole:
[tex]\[
\text{Volume of metal needed} = 2100 \pi - 756 \pi
\][/tex]
[tex]\[
= 2100 \pi - 756 \pi
\][/tex]
[tex]\[
= 1344 \pi \text{ cubic millimeters}
\][/tex]
From these calculations, we can identify the correct expressions representing the volume of metal needed:
1. [tex]\( 21 \pi (10)^2 - 21 \pi (6)^2 \)[/tex]
2. [tex]\( 2100 \pi - 756 \pi \)[/tex]
Both of these expressions are accurate formulations for the volume of metal required to make the pipe.
