Ten Examples

The solving process is much the same as solving problems with square units, except you have to cube the absolute exponential distance between the two prefixes. I will do the same examples as in the square unit tutorial, except as cubic problems.

Remember that cubic problems deal with volume.

**Example #1:** Convert 1.00 m^{3} to cubic centimeters.

**Solution:**

1) Here's the set up for the conversion:

??? cm ^{3}1.00 m ^{3}x ––––––– = the answer 1 m ^{3}Notice that m

^{3}cancels, leaving cm^{3}to be the unit on the answer.

2) Some discussion:

The absolute exponential distance between the base unit and centi- is 10^{2}. Sometimes a teacher may say the exponential distance is 2. Be aware that the 2 is an exponent, NOT that you will be multiplying something by 2. In a cubic situation, we must cube it to obtain 10^{6}. The reason we do this is because there are three centimeter measurements, one for each of the three sides of the cube.

3) Let me anticipate a problem:

You read the above and say "Wait, the cube of 2 is 8, isn't it?" You are right because (2)^{3}= 2 x 2 x 2 = 8, but we are dealing with an exponential value of 2, not the number 2. What gets cubed is 10^{2}, not 2. We would write (10^{2})^{3}= 10^{2 x 3}= 10^{6}.

4) Now, to finish the example:

10 ^{6}cm^{3}1.00 m ^{3}x ––––––– = 1.00 x 10 ^{6}cm^{3}1 m ^{3}You might also see it like this:

(1.00 m

^{3}) (10^{6}cm^{3}/ m^{3}) = 1.00 x 10^{6}cm^{3}

**Example #2:** Convert 1.00 km^{3} to cubic micrometers.

**Solution:**

1) Here's the problem done up as a conversion:

??? μm ^{3}1.00 km ^{3}x ––––––– = the answer 1 km ^{3}Notice that km

^{3}cancels, leaving μm^{3}to be the unit on the answer.

2) Some discussion:

Between kilo- and micro- is an absolute distance of 10^{9}. Our technique calls for the exponential value to be cubed, so (10^{9})^{3}= 10^{27}. The direction of change is going from the larger prefix to the smaller prefix, so the sign of the exponent in the answer will be positive.

3) Finish the example:

10 ^{27}μm^{3}1.00 km ^{3}x ––––––– = 1.00 x 10 ^{27}μm^{3}1 km ^{3}

The two examples above were going from the larger prefix to the smaller. That meant there were lots of the smaller prefix in one big prefix. Examples below will go the opposite direction. Ask yourself: how many of a larger prefix are there in one small prefix? Answer = less than one, so the sign on the exponent will be negative.

Having said that, watch out for a problem where there is an exponent already in the problem and the negative exponent change fails to overwhelm the exponent from the problem. In that case, there will be a positive exponent in the answer.

**Example #3:** Convert 4310 cubic centimeters to m^{3}.

**Solution:**

1 ^{3}m^{3}4310 cm ^{3}x ––––––– = 4.31 x 10¯ ^{3}m^{3}(10 ^{2})^{3}cm^{3}Notice how the 1 in the conversion fact is associated with the larger (the m

^{3}) of the two units. Larger here means in the sense that 1 m^{3}is larger than 1 cm^{3}.

**Example #4:** Convert 86.3 cubic centimeters to mm^{3}.

**Solution:**

86.3 cm^{3}x (10^{3}mm^{3}/ 1^{3}cm^{3}) = 8.63 x 10^{4}mm^{3}There are 10 mm in one cm, so 10 x 10 x 10 yields 10

^{3}mm^{3}.

**Example #5:** 5.94 x 10^{10} mm^{3}. Convert to dm^{3}. Then, convert to liters.

**Solution:**

1 dm ^{3}5.94 x 10 ^{10}mm^{3}x ––––––– = 5.94 x 10 ^{4}dm^{3}(100 mm) ^{3}Note how mm

^{3}goes into the denominator of the conversion factor, so as to cancel mm^{3}on the value to be converted.1 dm equals 100 mm, so 1 dm

^{3}= (100 mm)^{3}= 10^{6}mm^{3}Note how the 1 goes with dm

^{3}, the larger of the units when compared to 1 mm^{3}.5.94 x 10

^{4}dm^{3}= 5.94 x 10^{4}L because 1 dm^{3}equals 1 L.

## Convert L to m

^{3}

**Example #6:** A metal cube is 2.50 cm on a side. Calculate the volume in m^{3}.

**Solution:**

1) Convert 2.50 cm to meters:

(2.50 cm) (1 m / 100 cm) = 0.0250 m

2) Multiply to obtain the volume of the cube:

(0.0250 m) (0.0250 m) (0.0250 m) = 1.56 x 10¯^{5}m^{3}(to three sig figs)

3) Here's an alternate solution path:

(2.50 cm) (2.50 cm) (2.50 cm) = 15.625 cm^{3}(15.625 cm

^{3}) (1 m / 100 cm)^{3}= 15.625 x 10¯^{6}m^{3}15.625 x 10¯

^{6}m^{3}= 1.56 x 10¯^{5}m^{3}(to three sig figs)

**Example #7:** Convert 10.6 kg/m^{3} to g/cm^{3}

**Solution:**

1) Do the non-cubic conversion first, convert kg to g:

10 ^{3}g10.6 kg/m ^{3}x ––––– = 1.06 x 10 ^{4}g/m^{3}1 kg Note the use of exponential notation in the answer.

2) Do the cubic conversion next, convert meters cubed to centimeters cubed:

(1 m) ^{3}1.06 x 10 ^{4}g/m^{3}x ––––––– = 1.06 x 10¯ ^{2}g/cm^{3}(100 cm) ^{3}Note that 100 cubed is 10

^{6}. You could write the 100 cubed in the conversion as (10^{2})^{3}You could also write the answer as 0.0106 g/cm

^{3}.

3) Be careful to make double-sure you have everything in the right place. When I formatted this problem, I did so about 7:30 AM on a rainy Saturday. I had the conversion factor for the cubic conversion reversed. Needed MOAR COFFEE!!

**Example #8:** Convert the density of aluminum (2.70 g/cm^{3}) to kg/m^{3}.

**Solution:**

1) Here's the plan:

(a) convert g/cm^{3}to g/m^{3}(first conversion factor in the step just below)(b) convert g/m

^{3}to kg/m^{3}(second conversion factor in the step just below)

2) A little dimensional analysis:

(10 ^{2}cm)^{3}1 kg 2.70 g/cm ^{3}x ––––––– x ––––––– = 2.70 x 10 ^{3}kg/m^{3}(1 m) ^{3}1000 g

**Example #9:** Convert 409 cubic inches to liters using dimensional analysis.

**Solution:**

1) Here's the plan:

(a) convert in^{3}to cm^{3}(b) convert cm

^{3}to mL(c) convert mL to L

2) Here's the set up:

(2.54 cm) ^{3}1 mL 1 L 409 in ^{3}x ––––––––– x ––––– x ––––––– = 6.70 L (1 in) ^{3}1 cm ^{3}1000 mL

3) Comments:

(a) think of 1 in^{3}being a cube one inch by one inch by one inch. Since 1 inch equals 2.54 cm, one cubic inch equals a cube 2.54 cm by 2.54 cm by 2.54 cm.(b)1 cm

^{3}= 1 mL is a very handy conversion to remember.(c) Brian Wilson has a comment concerning the number 409.

**Example #10:** If a cube is 25.2 mm on a side, what is its volume in liters?

**Solution:**

1) Convert 25.2 mm to cm:

(25.2 mm) (1 cm / 10 mm) = 2.52 cm

2) Determine the volume of the cube:

(2.52 cm) (2.52 cm) (2.52 cm) = 16.003 cm^{3}

3) Convert cm^{3} to mL:

(16.003 cm^{3}) (1 mL / 1 cm^{3}) = 16.003 mL

4) Convert mL to L:

(16.003 mL) (1 L / 1000 mL) = 0.0160 L

**Bonus Example #1:** A mole of any ideal gas at STP occupies 22.414 L. How many molecules of hydrogen gas are in 22.414 mm^{3}?

**Solution:**

1) I'm going to build a dimensional analysis solution, one factor at a time. First, let's state the information from the problem:

1 mol ––––––– x 22.414 L I deliberately put the value that way because I know the final answer will be some number of molecules per cubic millimeter. The volume must go in the denominator.

2) We have to get rid of liters and go to some type of cubic measurement:

1 mol 1 L ––––––– x ––––– x 22.414 L 1 dm ^{3}Knowing the 1 L = 1 dm

^{3}is a handy conversion to remember.

3) Now, convert from cubic decimeters to cubic millimeters:

1 mol 1 L 1 dm ^{3}––––––– x ––––– x –––––––– x 22.414 L 1 dm ^{3}100 ^{3}mm^{3}for this:

100^{3}mm^{3}Remember that there are 100 mm in 1 dm, so there are:

100 mm x 100 mm x 100 mm in one dm^{3}

4) The last conversion is from moles to molecules:

1 mol 1 L 1 dm ^{3}6.022 x 10 ^{23}molecules––––––– x ––––– x –––––––– x ––––––––––––––––––– = 2.687 x 10 ^{16}molecules (to 4 sig figs)22.414 L 1 dm ^{3}100 ^{3}mm^{3}1 mol

**Bonus Example #2:** 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 m^{3}(to three sig figs)

4) Convert cubic meters to cubic feet:

(1.594216 m^{3}) (39.3701 inch / m)^{3}(1 ft / 12 in)^{3}= 56.3 ft^{3}(to three sig figs)Note that I used a cubic meter value with several extra digits, not the rounded off value of 1.59.