I am trying to get the Google Earth plug-in working on a page somewhere, but for now here is the KMZ file of a demonstration that shows the volume of the water in the world compared to the volume of the Earth in a set of aspiring spheres. There is another demonstration I also found here, that illustrates the amount of CO-2 emissions geographically in a cool spherical bar chart. Reading graphs is an important skill learned in a mathematics education, and it is interesting that new forms of information management tools, like Google Earth, are also enabling new types of graphs and visualizations.
Upon seeing the graphic above, I was first struck by how small the largest sphere of water was to the total size of the Earth. If this relationship is actually to scale, it shows how spreading a substance evenly over the surface of an object that is much larger its size occupies far less volume than the object itself. A question to this end would be how to figure out the average depth of water it would take to cover the Earth if it were evenly spread over its whole surfaceusing the numerical values from the demonstration.
The relationships between volumes of geometric objects can be very useful to intuit when making rather ordinary calculations. It can help you decide which pail you should use to get sand to fill up a sandbox, or the size of a pot to use to make Jambalaya when all the sausage and rice are done cooking. In understanding this relationship, it is helpful to know how the formulas for volumes and areas vary with quantities like length, height and radius.
We discussed how to determine volumes from areas when a shape has a uniform cross section int the post "Rapelling Down Fissures" -- there we learned that for some objects, their volume changed linearly with the height of the solid. So if we were planning on building a new water tower in our village when our population had almost doubled, we could build a water tower with the same circular cross section but just twice as high. If we decided to change the radius of our tank, however, and keep the height the same, would we need a new tank whose radius was twice as big as our previous one? What do you think?
To understand how much our radius must grow to accomodate the new required volume of water, it is helpful to remember that while the volume of our cylindrical water tank will vary linearly with its height, it will vary quadratically with the radius of its circular base. So, increasing the radius by a certain amount would actually lead to a change in volume corresponding to the radius squared, so we wouldn't need to adjust our radius in a one-to-one proportion to our required volume.
Q1: Use the numerical values from the graphic to determine the radii of each of the spheres representing different quantities of water.
Q2: What is the volume of saltwater on the Earth? What would be the radius of a sphere used to contain this much water?
Q3: Draw a graph of the volume of each of the quantities portrayed in the aspiring spheres verses the radii obtained in Q2. Does your graph make sense? Why or why not?
Upon seeing the graphic above, I was first struck by how small the largest sphere of water was to the total size of the Earth. If this relationship is actually to scale, it shows how spreading a substance evenly over the surface of an object that is much larger its size occupies far less volume than the object itself. A question to this end would be how to figure out the average depth of water it would take to cover the Earth if it were evenly spread over its whole surfaceusing the numerical values from the demonstration.
The relationships between volumes of geometric objects can be very useful to intuit when making rather ordinary calculations. It can help you decide which pail you should use to get sand to fill up a sandbox, or the size of a pot to use to make Jambalaya when all the sausage and rice are done cooking. In understanding this relationship, it is helpful to know how the formulas for volumes and areas vary with quantities like length, height and radius.
We discussed how to determine volumes from areas when a shape has a uniform cross section int the post "Rapelling Down Fissures" -- there we learned that for some objects, their volume changed linearly with the height of the solid. So if we were planning on building a new water tower in our village when our population had almost doubled, we could build a water tower with the same circular cross section but just twice as high. If we decided to change the radius of our tank, however, and keep the height the same, would we need a new tank whose radius was twice as big as our previous one? What do you think?
To understand how much our radius must grow to accomodate the new required volume of water, it is helpful to remember that while the volume of our cylindrical water tank will vary linearly with its height, it will vary quadratically with the radius of its circular base. So, increasing the radius by a certain amount would actually lead to a change in volume corresponding to the radius squared, so we wouldn't need to adjust our radius in a one-to-one proportion to our required volume.
QUIZ
Q1: Use the numerical values from the graphic to determine the radii of each of the spheres representing different quantities of water.
Q2: What is the volume of saltwater on the Earth? What would be the radius of a sphere used to contain this much water?
Q3: Draw a graph of the volume of each of the quantities portrayed in the aspiring spheres verses the radii obtained in Q2. Does your graph make sense? Why or why not?
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