To find the volume of metal needed to make the cylindrical pipe with a hole, we need to calculate the volumes of both the outer cylinder and the cylindrical hole, then subtract the volume of the hole from the volume of the pipe.
Given:
- The diameter of the cylindrical pipe is 20 mm, so its radius [tex]\( r_{\text{pipe}} \)[/tex] is 10 mm.
- The height [tex]\( h \)[/tex] of the cylindrical pipe is 21 mm.
- The radius of the cylindrical hole [tex]\( r_{\text{hole}} \)[/tex] is 6 mm.
First, let's determine the volume of the entire outer cylinder (the pipe without considering the hole):
[tex]\[ V_{\text{outer}} = \pi r_{\text{pipe}}^2 h \][/tex]
[tex]\[ V_{\text{outer}} = \pi (10)^2 (21) \][/tex]
[tex]\[ V_{\text{outer}} = 2,100 \pi \, \text{cubic millimeters} \][/tex]
Next, let's calculate the volume of the cylindrical hole:
[tex]\[ V_{\text{hole}} = \pi r_{\text{hole}}^2 h \][/tex]
[tex]\[ V_{\text{hole}} = \pi (6)^2 (21) \][/tex]
[tex]\[ V_{\text{hole}} = 756 \pi \, \text{cubic millimeters} \][/tex]
To find the volume of the metal needed, we subtract the volume of the hole from the volume of the outer cylinder:
[tex]\[ V_{\text{metal}} = V_{\text{outer}} - V_{\text{hole}} \][/tex]
[tex]\[ V_{\text{metal}} = 2,100 \pi - 756 \pi \][/tex]
[tex]\[ V_{\text{metal}} = 1,344 \pi \, \text{cubic millimeters} \][/tex]
Thus, the expressions that represent the volume of metal needed to make the pipe are:
[tex]\[ 21 \pi (10)^2 - 21 \pi (6)^2 \][/tex]
[tex]\[ 2,100 \pi - 756 \pi \][/tex]
Hence, the correct options are:
- [tex]\( 21 \pi (10)^2 - 21 \pi (6)^2 \)[/tex]
- [tex]\( 2,100 \pi - 756 \pi \)[/tex]
