Answer :
To determine the volume of a cylindrical capillary tube, we need to use the formula for the volume of a cylinder, which is given by:
[tex]\[ V = \pi r^2 h \][/tex]
where:
- [tex]\( V \)[/tex] is the volume,
- [tex]\( r \)[/tex] is the radius of the base of the cylinder,
- [tex]\( h \)[/tex] is the height of the cylinder,
- [tex]\( \pi \)[/tex] is a mathematical constant approximately equal to 3.14159.
Here are the steps to solve for the volume:
1. Measure the Diameter of the Tube:
- Use a pair of calipers or a ruler to measure the diameter of the capillary tube.
- Let's assume you measure the diameter to be [tex]\( d \)[/tex] centimeters.
2. Calculate the Radius:
- The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{d}{2} \][/tex]
- Record your radius measurement to the nearest 0.1 cm.
3. Measure the Height of the Tube:
- Using a ruler or another measurement tool, measure the height [tex]\( h \)[/tex] of the capillary tube.
- Though in some contexts, a default height may be given, in this scenario you should measure it yourself.
4. Calculate the Volume:
- Plug the values of [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the volume formula.
- Calculate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \left(\frac{d}{2}\right)^2 \][/tex]
- Multiply by [tex]\( \pi \)[/tex] and then by the height [tex]\( h \)[/tex]:
[tex]\[ V = \pi \left(\frac{d}{2}\right)^2 h \][/tex]
### Example Calculation:
Suppose you measured the diameter [tex]\( d \)[/tex] to be 2.0 cm, and the height [tex]\( h \)[/tex] to be 10.0 cm.
1. Calculate the radius:
[tex]\[ r = \frac{2.0}{2} = 1.0 \, \text{cm} \][/tex]
2. Calculate the volume:
[tex]\[ V = \pi (1.0)^2 \times 10.0 \][/tex]
[tex]\[ V = \pi \times 1.0 \times 10.0 \][/tex]
[tex]\[ V = 10.0 \pi \][/tex]
[tex]\[ V \approx 10.0 \times 3.14159 \][/tex]
[tex]\[ V \approx 31.4159 \, \text{cm}^3 \][/tex]
So, the volume of the cylindrical capillary tube would be approximately 31.4159 cubic centimeters.
[tex]\[ V = \pi r^2 h \][/tex]
where:
- [tex]\( V \)[/tex] is the volume,
- [tex]\( r \)[/tex] is the radius of the base of the cylinder,
- [tex]\( h \)[/tex] is the height of the cylinder,
- [tex]\( \pi \)[/tex] is a mathematical constant approximately equal to 3.14159.
Here are the steps to solve for the volume:
1. Measure the Diameter of the Tube:
- Use a pair of calipers or a ruler to measure the diameter of the capillary tube.
- Let's assume you measure the diameter to be [tex]\( d \)[/tex] centimeters.
2. Calculate the Radius:
- The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{d}{2} \][/tex]
- Record your radius measurement to the nearest 0.1 cm.
3. Measure the Height of the Tube:
- Using a ruler or another measurement tool, measure the height [tex]\( h \)[/tex] of the capillary tube.
- Though in some contexts, a default height may be given, in this scenario you should measure it yourself.
4. Calculate the Volume:
- Plug the values of [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the volume formula.
- Calculate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \left(\frac{d}{2}\right)^2 \][/tex]
- Multiply by [tex]\( \pi \)[/tex] and then by the height [tex]\( h \)[/tex]:
[tex]\[ V = \pi \left(\frac{d}{2}\right)^2 h \][/tex]
### Example Calculation:
Suppose you measured the diameter [tex]\( d \)[/tex] to be 2.0 cm, and the height [tex]\( h \)[/tex] to be 10.0 cm.
1. Calculate the radius:
[tex]\[ r = \frac{2.0}{2} = 1.0 \, \text{cm} \][/tex]
2. Calculate the volume:
[tex]\[ V = \pi (1.0)^2 \times 10.0 \][/tex]
[tex]\[ V = \pi \times 1.0 \times 10.0 \][/tex]
[tex]\[ V = 10.0 \pi \][/tex]
[tex]\[ V \approx 10.0 \times 3.14159 \][/tex]
[tex]\[ V \approx 31.4159 \, \text{cm}^3 \][/tex]
So, the volume of the cylindrical capillary tube would be approximately 31.4159 cubic centimeters.
