004. Units of volume measure

 

Question: `q001. There are 17 questions in this document.

 

How many cubic centimeters of fluid would require to fill a cubic container 10 cm on a side?

 

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Confidence Assessment:

 

Given Solution:

 

`aThe volume of the container is 10 cm * 10 cm * 10 cm = 1000 cm^3. So it would take 1000 cubic centimeters of fluid to fill the container.

 

 

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Question: `q002. How many cubes each 10 cm on a side would it take to build a solid cube one meter on a side?

 

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Confidence Assessment:

 

Given Solution:

 

`aIt takes ten 10 cm cubes laid side by side to make a row 1 meter long or a tower 1 meter high. It should therefore be clear that the large cube could be built using 10 layers, each consisting of 10 rows of 10 small cubes. This would require 10 * 10 * 10 = 1000 of the smaller cubes.

 

 

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Question: `q003. How many square tiles each one meter on each side would it take to cover a square one km on the side?

 

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Confidence Assessment:

 

Given Solution:

 

`aIt takes 1000 meters to make a kilometer (km). To cover a square 1 km on a side would take 1000 rows each with 1000 such tiles to cover 1 square km. It therefore would take 1000 * 1000 = 1,000,000 squares each 1 m on a side to cover a square one km on a side.

 

We can also calculate this formally. Since 1 km = 1000 meters, a square km is (1 km)^2 = (1000 m)^2 = 1,000,000 m^2.

 

 

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Question: `q004. How many cubic centimeters are there in a liter?

 

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Confidence Assessment:

 

Given Solution:

 

`aA liter is the volume of a cube 10 cm on a side. Such a cube has volume 10 cm * 10 cm * 10 cm = 1000 cm^3. There are thus 1000 cubic centimeters in a liter.

 

 

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Question: `q005. How many liters are there in a cubic meter?

 

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Confidence Assessment:

 

Given Solution:

 

`aA liter is the volume of a cube 10 cm on a side. It would take 10 layers each of 10 rows each of 10 such cubes to fill a cube 1 meter on a side. There are thus 10 * 10 * 10 = 1000 liters in a cubic meter.

 

 

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Question: `q006. How many cm^3 are there in a cubic meter?

 

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Confidence Assessment:

 

Given Solution:

 

`aThere are 1000 cm^3 in a liter and 1000 liters in a m^3, so there are 1000 * 1000 = 1,000,000 cm^3 in a m^3.

 

It's important to understand the 'chain' of units in the previous problem, from cm^3 to liters to m^3. However another way to get the desired result is also important:

 

There are 100 cm in a meter, so 1 m^3 = (1 m)^3 = (100 cm)^3 = 1,000,000 cm^3.

 

STUDENT COMMENT

 

It took me a while to decipher this one out, but I finally connected the liters to cm^3 and m^3. I should have calculated it by just converting units, it would have been easier.

 

INSTRUCTOR RESPONSE

 

The point isn't just conversion. There are two points to understanding the picture. One is economy of memory: it's easier to remember the picture than the conversion factors, which can easily be confused. The other is conceptual/visual: the picture gives you a deeper understanding of the units.

In the long run it's easier to remember that a liter is a 10-cm cube, and a cubic meter is a 100-cm cube.

Once you get this image in your mind, it's obvious how 10 layers of 10 rows of 10 one-cm cubes forms a liter, and 10 layers of 10 rows of 10 one-liter cubes forms a cubic meter.

Once you understand this, rather than having a meaningless conversion number you have a picture that not only gives you the conversion, but can be used to visualize the meanings of the units and how they are applied to a variety of problems and situations.

 

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Question: `q007. If a liter of water has a mass of 1 kg the what is the mass of a cubic meter of water?

 

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Confidence Assessment:

 

Given Solution:

 

Since there are 1000 liters in a cubic meter, the mass of a cubic meter of water will be 1000 kg. This is a little over a ton.

 

 

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Question: `q008. What is the mass of a cubic km of water?

 

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Confidence Assessment:

 

Given Solution:

 

 

`aA cubic meter of water has a mass of 1000 kg. A cubic km is (1000 m)^3 = 1,000,000,000 m^3, so a cubic km will have a mass of 1,000,000,000 m^3 * 1000 kg / m^3 = 1,000,000,000,000 kg.

 

In scientific notation we would say that 1 m^3 has a mass of 10^3 kg, a cubic km is (10^3 m)^3 = 10^9 m^3, so a cubic km has mass (10^9 m^3) * 1000 kg / m^3 = 10^12 kg.

 

STUDENT QUESTION

 

I don’t understand why you multiplied the 1,000,000,000 m^3 by 1000 km/m^3. I also don’t understand where the (1000m)^3 came from. I thought I had this problem but it stumped me. It is probably something really simple that I am missing.  ???

INSTRUCTOR RESPONSE

 

A km is 1000 meters, but a cubic km is a cube 1000 meters on a side. It would take 1000 m^3 just to make a single row of 1-m cubes 1000 meters long, and you would hardly have begun constructing a cubic kilometer. You would need 1000 such rows just to cover a 1-km square 1 meter deep, and 1000 equal layers to build a cube 1 km high.

Each layer would require 1000 * 1000 cubic meters, and 1000 layers would require 1000 times this many 1-meter cubes.

 

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Question: `q009. If each of 5 billion people drink two liters of water per day then how long would it take these people to drink a cubic km of water?

 

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Confidence Assessment:

 

Given Solution:

 

`a5 billion people drinking 2 liters per day would consume 10 billion, or 10,000,000,000, or 10^10 liters per day.

 

A cubic km is (10^3 m)^3 = 10^9 m^3 and each m^3 is 1000 liters, so a cubic km is 10^9 m^3 * 10^3 liters / m^3 = 10^12 liters, or 1,000,000,000,000 liters.

 

At 10^10 liters per day the time required to consume a cubic km would be

 

time to consume 1 km^3 = 10^12 liters / (10^10 liters / day) = 10^2 days, or 100 days.

 

This calculation could also be written out:

 

1,000,000,000,000 liters / (10,000,000,000 liters / day) = 100 days.

 

STUDENT COMMENT

 

There came to be too many conversions for me to keep in memory all of the conversions about and how they work together, so I
had to write out all of the conversions next to each other and multiply them all out, even if I could have made some
shortcuts, such as the numbers of liters in a cubic meter.

INSTRUCTOR RESPONSE

 

You can easily visualize a 1-cm cube, a 10-cm cube and a 1-m cube. You should be able to visualize how each is built up from 1000 of the previous. If you understand the model and make it tangible there is no need to memorize anything, and you will have a significant measure of protections against errors with these quantities.

By understanding the meaning of the prefix 'kilo' it is easy enough to then relate these units to the somewhat less tangible cubic kilometer.

 

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Question: `q010. The radius of the Earth is approximately 6400 kilometers. What is the surface area of the Earth? If the surface of the Earth was covered to a depth of 2 km with water that what would be the approximate volume of all this water?

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Confidence Assessment:

 

Given Solution:

 

`aThe surface area would be

 

A = 4 pi r^2 = 4 pi ( 6400 km)^2 = 510,000,000 km^2.

 

A flat area of 510,000,000 km^2 covered to a depth of 2 km would indicate a volume of

 

V = A * h = 510,000,000 km^2 * 2 km = 1,020,000,000 km^3.

 

However the Earth's surface is curved, not flat. The outside of the 2 km covering of water would have a radius 2 km greater than that of the Earth, and therefore a greater surface area. But a difference of 2 km in 6400 km will change the area by only a fraction of one percent, so the rounded result 1,020,000,000,000 km^3 would still be accurate.

 

STUDENT COMMENT

 

I thought that in general pi was always supposed to be expressed as pi when not asked for an approximate value so in the
first part of the problem I didn’t calculate pi. For the second part of the question I assumed approximate meant calculate
pi into the equation which would still be a less precise answer so I did not round any further. ???Should I have estimated
more than I did???

INSTRUCTOR RESPONSE

 

The given information says 'approximately 6400 km'.

Your result, 163,840,000pi km^2, is perfectly fine.

However most people aren't going to recognize 163,840,000 as 4 times the square of 6400 (unlike a result like 36 pi (easily enough seen as either 6^2 * pi, or 4 * 3^2 * pi)). Since the given information is accurate to only a couple of significant figures, there's no real advantage in the multiple-of-pi expression.

In the given solution the results are generally expressed to 2 significant figures, consistent with the 2 significant figures in the given 6400 km radius.

 

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Question: `q011. Summary Question 1: How can we visualize the number of cubic centimeters in a liter?

 

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Confidence Assessment:

 

Given Solution:

 

Since a liter is a cube 10 cm on a side, we visualize 10 layers each of 10 rows each of 10 one-centimeter cubes, for a total of 1000 1-cm cubes. There are 1000 cubic cm in a liter.

 

 

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Question: `q012. Summary Question 2: How can we visualize the number of liters in a cubic meter?

 

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Confidence Assessment:

 

Given Solution:

 

Since a liter is a cube 10 cm on a side, we need 10 such cubes to span 1 meter. So we visualize 10 layers each of 10 rows each of 10 ten-centimeter cubes, for a total of 1000 10-cm cubes. Again each 10-cm cube is a liter, so there are 1000 liters in a cubic meter.

 

 

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Question: `q013. Summary Question 3: How can we calculate the number of cubic centimeters in a cubic meter?

 

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Confidence Assessment:

 

Given Solution:

 

`aOne way is to know that there are 1000 liters in a cubic meters, and 1000 cubic centimeters in a cubic meter, giving us 1000 * 1000 = 1,000,000 cubic centimeters in a cubic meter. Another is to know that it takes 100 cm to make a meter, so that a cubic meter is (100 cm)^3 = 1,000,000 cm^3.

 

 

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Question: `q014. Summary Question 4: There are 1000 meters in a kilometer. So why aren't there 1000 cubic meters in a cubic kilometer? Or are there?

 

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Confidence Assessment:

 

Given Solution:

 

`aA cubic kilometer is a cube 1000 meters on a side, which would require 1000 layers each of 1000 rows each of 1000 one-meter cubes to fill. So there are 1000 * 1000 * 1000 = 1,000,000,000 cubic meters in a cubic kilometer.

 

Alternatively, (1 km)^3 = (10^3 m)^3 = 10^9 m^3, not 1000 m^3.

 

STUDENT ANSWER to question:

 

Because a cubic kilometer is cubed. A regular kilometer is not going to contain as much as a cubic kilometer.

INSTRUCTOR RESPONSE

 

Kilometers and cubic kilometers don't measure the same sort of thing, so they can't be compared at all.

Kilometers measure distance, how far it is between two points.

Cubic kilometers measure volume, how much space there is inside of something (there is space, though not necessarily empty space, inside of any container or any 3-dimensional region, whether it's full of other stuff or not. If it's full of other stuff then we wouldn't say that it's 'empty space' or 'available space', but the amount of space inside is the same either way).
 

 

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Question: `q015. A micron is a thousandth of a millimeter.

 

A certain pollen grain is an approximate cube 10 microns on a side. 

 

In as many ways as possible, without using a formula, reason out the volume of the pollen grain in cubic microns.

 

In as many ways as possible, again without using formulas, reason out how many such pollen grains could fit in a cube one centimeter on a side.

Your solution:

 

 

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Question: `q016. If the surface area of a human body is 2 square meters, and if it is covered uniformly with a layer of perspiration 100 microns thick, then what are the volume and the mass of that perspiration layer?

 

Your solution:

 

 

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Question: `q017. Explain how you have organized your knowledge of the principles illustrated by the exercises in this assignment.

 

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