#$&* course Phy 232 004. Units of volume measure
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Given Solution: `aThe volume of the container is 10 cm * 10 cm * 10 cm = 1000 cm^3. So it would take 1000 cubic centimeters of fluid to fill the container. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q002. How many cubes each 10 cm on a side would it take to build a solid cube one meter on a side? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To make up the larger cube out of smaller cubes, each one meter on a side, it would take 10^3 = 1,000 smaller cubes. This is because on a given side of the bigger cube, it would take 10 smaller cubes per row, and another 10 to make the columns. Also, another 10 smaller cubes come into the equation because the bigger cube is 10 smaller cubes thick. After putting all of these things together, we get 10^3. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q003. How many square tiles each one meter on each side would it take to cover a square one km on the side? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To cover a square that’s one km on its side, we have to first convert the tiles from meters to kilometers. 1 kilometer = 1,000 meters. Since it would take 1,000 of these tiles to cover a row, and 1,000 rows were also needed, we come up with the equation of 1,000*1,000 = 1,000,000 small tiles. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aIt takes 1000 meters to make a kilometer (km). To cover a square 1 km on a side would take 1000 rows each with 1000 such tiles to cover 1 square km. It therefore would take 1000 * 1000 = 1,000,000 squares each 1 m on a side to cover a square one km on a side. We can also calculate this formally. Since 1 km = 1000 meters, a square km is (1 km)^2 = (1000 m)^2 = 1,000,000 m^2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q004. How many cubic centimeters are there in a liter? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find how many cubic centimeters are in a liter, we need to find how many cubic centimeters are in a 10X10X10 cube. We need to know this because such cube is equal to a liter in volume. Therefore, the amount of cubic centimeters in a 10X10X10 cube is the same number for how many of them there are in a liter. Therefore, 10^3 = 1,000 cm^3, which means that there are 1,000 cubic centimeters in a liter. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q005. How many liters are there in a cubic meter? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We know that a 10X10X10 cube’s volume is equal to one liter. Knowing that we can find out how many liters are in a cubic centimeter. To make up a side of a cube that has one meter on its sides, again we must do the calculation 10^3 = 1,000 liters in a cubic meter. This is because there needs to be 10 rows of 10 cubes and they are 10 deep, giving us the answer. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aA liter is the volume of a cube 10 cm on a side. It would take 10 layers each of 10 rows each of 10 such cubes to fill a cube 1 meter on a side. There are thus 10 * 10 * 10 = 1000 liters in a cubic meter. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q006. How many cm^3 are there in a cubic meter? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If there are 1,000 cm^3 in a liter, and 1,000 liters in a m^3, then there are (1,000*1,000) cm^3 in a m^3. This calculation comes out to be 1,000,000 cm^3 in a m^3. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q007. If a liter of water has a mass of 1 kg the what is the mass of a cubic meter of water? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We know the relationship that one liter equals 1,000 cubic meters; therefore if one liter weighs 1 kg, then a cubic meter would weigh 1,000 kg. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Since there are 1000 liters in a cubic meter, the mass of a cubic meter of water will be 1000 kg. This is a little over a ton. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q008. What is the mass of a cubic km of water? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We can find that a km^3 is equivalent to 1,000,000,000 m^3 because it takes (1,000^3) m to make a km^3. And with one m^3 weighing in at 1,000 kg, we can calculate the weight of a km^3 of water. This is done by, 1,000,000,000 * 1,000 = 1,000,000,000,000 kg. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: First, we must calculate how much water per day is being consumed by the 5 billion people. If each person drinks 2 liters per day, then the total consumption per day would be 2*5,000,000,000 = 10,000,000,000 liters (10 billion). Next, we know from previous conversions that it would take 1,000,000,000,000 liters to make up a km^3. Therefore, with a consumption rate known and a total amount known in the same units, we can calculate how long it takes for the people to drink a km^3 of water. We must do the total amount needed divided by the daily consumption to find the number of days it takes to consume that much water. Therefore, (1,000,000,000,000 / 10,000,000,000 = 100 days. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To calculate the surface area of the Earth, we can use the surface area formula for a sphere, which is 4*pi*(r^2). Plugging in the given radius for the Earth we get, 4*pi*(6,400^2) = 163,840,00*pi km^2 or 514,457,600 km^2. Using that area to determine the volume is done by the formula, volume = area*height(depth). Since the depth is 2, we calculate the volume as 514,457,600 km^2 * 2 km = 1,028,915,200 km^3 as the Earth’s volume plus the added water. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q011. Summary Question 1: How can we visualize the number of cubic centimeters in a liter? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: A liter is described at a cube with length 10 cm on each side. We know that there has to be 10 layers, each with 10 rows containing 10 smaller cubes, each with a side length of one cubic centimeter. Therefore, to calculate the total number of cubic centimeters in a liter, we say that 10*10*10 = 1,000 cubic centimeters make up this larger cube which is equivalent to a liter. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Since a liter is a cube 10 cm on a side, we visualize 10 layers each of 10 rows each of 10 one-centimeter cubes, for a total of 1000 1-cm cubes. There are 1000 cubic cm in a liter. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q012. Summary Question 2: How can we visualize the number of liters in a cubic meter? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The same principle applies here as to the previous problem dealing with the cubic centimeters and the liter. We know that the liter is a cube with side lengths of 10 cm on each side. We also know that there are 10 of these cubes that make up 1 meter. The same process of 10 layers with 10 rows with 10 each in them occurs, giving us a total of 10^3 = 1,000 L in a m^3. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q013. Summary Question 3: How can we calculate the number of cubic centimeters in a cubic meter? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given the two ratios from cubic centimeter to liter, and from liter to cubic meter, we can calculate how many cubic centimeters that are in one m^3. Since there is 1,000 cubic centimeters in a liter, and 1,000 liters in a cubic meter, then there is 1,000*1,000 = 1,000,000 cubic centimeters in a cubic meter. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aOne way is to know that there are 1000 liters in a cubic meters, and 1000 cubic centimeters in a cubic meter, giving us 1000 * 1000 = 1,000,000 cubic centimeters in a cubic meter. Another is to know that it takes 100 cm to make a meter, so that a cubic meter is (100 cm)^3 = 1,000,000 cm^3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q014. Summary Question 4: There are 1000 meters in a kilometer. So why aren't there 1000 cubic meters in a cubic kilometer? Or are there? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: A cubic kilometer is represented by a cube with 1,000 meter smaller cubes on each of its sides. Therefore taking into account the layer and rows and how many there were in each row, we come up with the calculation 1,000*1,000*1,000 or 1,000^3 and get 1,000,000,000 cubic meters in a cubic kilometer. A kilometer would not equal as much as a km^3. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This pollen cube has 10 microns one each side. Therefore to find the volume, we know that there is 10 micron layers deep, to make up the cube, along with 10 rows on each side made up of 10 microns a row. Therefore, once you multiply out all the numbers included, you find the volume of the pollen cube. The same principle is applied to a cube using cubic centimeters instead of cubic microns. Multiplying the total number of rows on each side of the cube by the total number of cubes in each row, and also by the total number of cubes deep, gives you the final volume of the entire cube that is made up of these smaller cubes that have side lengths of cubic centimeters.