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A cylindrical metal pipe has a diameter of 20 millimeters and a height of 21 millimeters. A cylindrical hole cut out of the center has a radius of 6 millimeters.

Which expressions represent the volume of metal needed, in cubic millimeters, to make the pipe? Select two options.

A. [tex]21 \pi(10)^2 - 21 \pi(6)^2[/tex]
B. [tex]\pi(20)^2(21) - \pi(6)^2[/tex]
C. [tex]2,100 \pi - 756 \pi[/tex]
D. [tex]7,644 \pi[/tex]
E. [tex]1,344[/tex]

Sagot :

GinnyAnswer
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]
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