Metric Cubic Unit Conversion
Problems #1-10

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Problem #1: Convert 0.500 cubic feet to liters

Solution:

1) Let's set up the dimensional analysis first and then comment on it:

    (12 inch)3   (2.54 cm)3   1 mL   1 L  
0.5 ft3  x  –––––––  x  –––––––  x  –––––  x  –––––––  =  14.2 L
    1 ft3   (1 in)3   1 cm3   1000 mL  

2) The conversions:

first ---> convert cubic feet to cubic inches
second ---> converts in3 to cubic centimeters
third ---> converts cm3 to mL
fourth ---> converts mL to L

3) Comment:

Note the (12 inch)3. You must remember to "distribute" the cube. While it is true that 12 inches equals 1 foot, you have to remember that 12 in3 DOES NOT equal 1 ft3. You have to cube the 12, as in:
1 ft3 = (12 in) (12 in) (12 in) = 1728 in3

The number 12 AND the unit inch both get cubed.


Problem #2: Convert 0.500 cubic feet to dm3

Solution:

1) The dimensional analysis set up:

    (12 inch)3   (2.54 cm)3   (1 dm)3  
0.5 ft3  x  –––––––  x  –––––––  x  –––––––  =  14.2 dm3
    1 ft3   (1 in)3   (10 cm)3  

2) The conversions:

first ---> convert cubic feet to cubic inches
second ---> converts in3 to cubic centimeters
third ---> converts cm3 to dm3

3) Comment:

There are 10 cm in 1 dm. Remember, deci- means 0.1 while centi- means 0.01. There are 1000 cm3 in 1 dm3, just like there are 1000 mL in 1 L.

Problem #3: Convert the density of gold, 19.3 g/cm3, to kg/L.

Solution:

1) Convert grams to kg:

19.3 g/cm3 times (1 kg / 1000 g) = 0.0193 kg/cm3

2) Convert cm3 to mL:

0.0193 kg/cm3 times (1 mL / 1 cm3) = 0.0193 kg/mL

3) Convert mL to L:

0.0193 kg/mL times (1000 mL / L) = 19.3 kg/L

4) Often, a teacher will want you to write out the conversions in one long line:

19.3 g/cm3 x (1 kg / 1000 g) x (1 mL / 1 cm3) x (1000 mL / L) = 19.3 kg/L

Problem #4: Change 1.0 kg/cm3 to g/mm3

Solution:

(1.0 kg / cm3) x (1000 g / kg) = 1000 g/cm3

(1000 g / cm3) x (1 cm3 / 1000 mm3) = 1.0 g / mm3

Remember, 10 mm in one cm, so 10 mm x 10 mm x 10 mm equals 1000 mm3 in one cm3.

Teachers like these conversions where you start with 1 and end with 1. Students don't expect it and think they have done something wrong. The video has another example.


Convert kg/m3 to g/L


Problem #5: Convert 25.0 mL to mm3 using dimensional analysis.

Solution via cm3:

  1 cm3   103 mm3  
25.0 mL x  –––––  x  –––––––  = 2.50 x 104 mm3
  1 mL   1 cm3  

The 1 mL equals 1 cm3 conversion is very handy.

There are 10 mm in every cm, so 10 cubed is 103.

Solution via dm3:

  1 L   1 dm3   106 mm3  
25.0 mL x  –––––  x  –––––  x  –––––––  = 2.50 x 104 mm3
  103 mL   1 L   1 dm3  

The 1 L equals 1 dm3 conversion is very handy.

There are 100 mm in every dm, so 100 cubed is 106


Problem #6: Convert 5.51 g/cm3 to lb/ft3.

Solution:

1) Do the non-cubic conversion first:

  1 lb  
5.51 g/cm3 x  ––––––  = the answer
  453.6 g  

I'm not going to work it out yet, I simply wanted to show the above conversion. I'll leave the working out until the cm3 to ft3 is added in.

2) We can look up a direct conversion from cm3 to ft3 (you may, I will not bother). The usual technique (which is what I follow) is to go via common conversions, ones that teachers tend to have their students memorize. In this specific case, we would go from cm3 to cubic inch (since 1 inch exactly equals 2.54 cm) and then from in3 to cubic foot (because 1 foot exactly equals 12 inches).

  1 lb   (2.54 cm)3   (12 in)3  
5.51 g/cm3 x  ––––––  x  ––––––––  x  ––––––––  = 344 lb/ft3
  453.6 g   (1 in)3   (1 ft)3  

Notice the style I used for the cubic units: (1 ft)3, (2.54 cm)3, and so on. Sometimes, you see it done without the parentheses: 1 ft3 or 2.543 cm3 might be what you woud see. Notice how the cube on the numeral 1 is eliminated.

5.51 g/cm3 is the density of the Earth.


Problem #7: How many 1 cm cubes would it take to construct a cube that is 4 cm on edge?

Solution:

1) The formula for volume of a cube is:

V = l*w*h

2) Insert 4 cm for each of the three dimensions:

V = 4 cm times 4 cm times 4 cm = 64 cm3

Sixty-four 1 cm cubes would be required.


Problem #8: Convert 6.230 x 10¯2 kg/mm3 to g/L

Solution:

1) You have to do a kg to g conversion and then take mm3 to cm3 to mL to L. Here it is:

0.06230 kg   1000 g   (10 mm)3   (1 cm)3   1000 mL  
–––––––––  x  –––––  x  –––––––  x  –––––––  x  –––––––  =  6.230 x 107 g/L
mm3   1 kg   (1 cm)3   1 mL   1 L  

2) Notice that kg/mm is in the form of a density. When I formatted the above problem, I thought to explore a 'calculate the density of a black hole' problem, but I decided against it. However, here is a density conversion problem that might interest you. If you scroll up to the top of the file, you'll find some discussion about the density of a black hole.


Problem #9: If a box measures 3.00 cm wide, 50.5 mm long and 0.520 m high, what is its volume in cubic feet?

Solution:

1) One of the common pathways from metric units to English units is 2.54 cm = 1 inch. Let's take each measurement to cm and then calculate the volume in cm3:

(50.5 mm) (1 cm / 10 mm) = 5.05 cm
(0.520 m) (100 cm / 1 m) = 520. cm

volume ---> (3.00 cm) (5.05 cm) (520. cm) = 7878 cm3

2) Convert from cm3 to cubic inches:

(7878 cm3) (1 inch / 2.54 cm)3 = 480.745 in3

Notice how I cubed the entire conversion factor rather than cubing them individually, as in (1 in)3 / (2.54 cm)3

3) Convert from cubic inches to cubic feet:

(480.745 in3) (1 foot / 12 inches)3 = 0.278 ft3 (to three sig figs)

Problem #10: A metal cube has the following dimensions: width is 5.20 inch, length is 6.00 cm and height is 1.00 furlong. What is the volume of the cube in cubic meters? In cubic yards?

Solution:

1) Let's convert each measurement to meters. All we have to do then is multiply together to get cubic meters:

(5.20 inch) (1 m / 39.3701 inches) = 0.13208 m
(6.00 cm) (1 m / 100 cm) = 0.0600 m

2) Furlong is a bit more involved:

    220 yard   3 ft   12 inches   2.54 cm   1 m  
1.00 furlong  x  –––––––  x  –––––––  x  –––––––  x  –––––––  x  –––––––  =  201.168 m
    1 furlong   1 yd   1 ft   1 in   100 cm  

3) Just a reminder to be careful. When I set up the conversion for furlong to meters, I swapped the 2.54 and the 1. I know this stuff fairly well and still fall victim to the occasional error. Check your work! Here's the volume in cubic meters:

(0.13208 m) (0.0600 m) (201.168 m) = 1.59 m3 (to three sig figs)

4) Convert cubic meters to cubic feet:

(1.594216 m3) (39.3701 inch / m)3 (1 ft / 12 in)3 = 56.3 ft3 (to three sig figs)

Note that I used a cubic meter value with several extra digits, not the rounded off value of 1.59.


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