To find the volume of metal needed to make a cylindrical pipe with a diameter of 20 millimeters (which gives an outer radius of 10 millimeters) and a height of 21 millimeters, with an inner cylindrical hole of radius 6 millimeters, we can follow these steps:
1. Calculate the volume of the outer cylinder (the pipe's outer surface):
- The radius of the outer cylinder [tex]\( r_{\text{outer}} \)[/tex] is 10 mm.
- The height [tex]\( h \)[/tex] of the cylinder is 21 mm.
[tex]\[ \text{Volume}_{\text{outer}} = \pi \times r_{\text{outer}}^2 \times h \][/tex]
[tex]\[ \text{Volume}_{\text{outer}} = \pi \times (10)^2 \times 21 \][/tex]
[tex]\[ \text{Volume}_{\text{outer}} = \pi \times 100 \times 21 \][/tex]
[tex]\[ \text{Volume}_{\text{outer}} = 2100 \pi \, \text{mm}^3 \][/tex]
2. Calculate the volume of the inner cylindrical hole:
- The radius of the inner cylinder [tex]\( r_{\text{inner}} \)[/tex] is 6 mm.
- The height [tex]\( h \)[/tex] remains the same, 21 mm.
[tex]\[ \text{Volume}_{\text{inner}} = \pi \times r_{\text{inner}}^2 \times h \][/tex]
[tex]\[ \text{Volume}_{\text{inner}} = \pi \times (6)^2 \times 21 \][/tex]
[tex]\[ \text{Volume}_{\text{inner}} = \pi \times 36 \times 21 \][/tex]
[tex]\[ \text{Volume}_{\text{inner}} = 756 \pi \, \text{mm}^3 \][/tex]
3. Calculate the volume of metal needed:
- This is found by taking the difference between the volume of the outer cylinder and the volume of the inner hole.
[tex]\[ \text{Volume}_{\text{metal}} = \text{Volume}_{\text{outer}} - \text{Volume}_{\text{inner}} \][/tex]
[tex]\[ \text{Volume}_{\text{metal}} = 2100 \pi - 756 \pi \][/tex]
[tex]\[ \text{Volume}_{\text{metal}} = (2100 - 756) \pi \][/tex]
[tex]\[ \text{Volume}_{\text{metal}} = 1344 \pi \, \text{mm}^3 \][/tex]
Given this, we can identify the correct expressions from the options provided:
1. [tex]\( 21 \pi (10)^2 - 21 \pi (6)^2 \)[/tex] (This simplifies to [tex]\( 2100 \pi - 756 \pi \)[/tex])
2. [tex]\( 2100 \pi - 756 \pi \)[/tex]
Both [tex]\( 21 \pi(10)^2 - 21 \pi(6)^2 \)[/tex] and [tex]\( 2100 \pi - 756 \pi \)[/tex] represent the correct expressions for the volume of metal needed to make the pipe.
