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My interest in this lies in my previous experience as a high school chemistry instructor. In some of my suggested methods for solving this problem include the physical science concept of density. Density is the measure of the amount of mass in a unit volume of substance. That is,
In addition, the ideas of an upper bound and a lower bound to the solutions can be investigated. Thus, I will look at ways in which an algebra II class can be team taught with a chemistry or physical science class to give the students a practical application to their math.
We can estimate the volume by knowing the following relationships: the number of ounces in a liter, the number of milliliters in a liter, the number of inches in a centimeter. I am old enough to remember Pepsi-Cola being sold in 64 fl. oz. glass bottles. These were changed sometime in the 1970s to the current 2 L, non-ecologically correct, plastic bottles. So my first calculation is from the approximation 2 litres approximately equals to 64 oz. So,
12 oz. * (2 L/64 oz.) * (1000 mL/1 L) ~ 375.0 mL = 375.0 cm^3 * (1 in/2.54 cm)^3 ~ 22.9 in^3
A refinement of the above procedure will be to use the equivalence that is on the label of a 2 L bottle of soda. That is, 2 L ~ 67.6 oz. So,
12 oz. * (2 L/67.6 oz.) * (1000 mL/1 L) ~ 355.0 mL = 355.0 cm^3 * (1 in/2.54 cm)^3 ~ 21.7 in^3
Another refinement is to use the equivalence that is on the label of a 20 oz. bottle of soda. That is, 591 mL ~ 20 oz. So,
12 oz. * (591 mL/20 oz.) ~ 354.6 mL = 354.6 cm^3 * (1 in/2.54 cm)^3 ~ 21.6 in^3
However, these are underestimates of the volume of the 12 oz. can. The above calculations are the volume of the soda but they do not include the volume of the carbon dioxide gas above the soda solution that is used to keep the aqueous solution carbonated. So a better method will be to measure the dimensions of the can directly. But since the can is not a perfect cylinder, we can only estimate the upper bound and the lower bound of the actual volume.
Place the can on a flat surface. Place a ruler on the top of the can. The height of the can is the distance measured from the surface to the ruler. I obtained 4.5 inches. The only other measure needed is the radius. We can obtain this from the diameter. To get an accurate measure of the diameter, take a piece of string and wrap it around the can using the ruler to make sure the string is parallel with the flat surface. Mark and measure the length of the string. To find the radius use the formula for the circumference of the circle: C = 2 * pi * r. I measured the circumference to be 8-5/8 inches. Thus, the radius, r = C/2pi = 1.37 inches. Finally the volume of the can, V = pi * r^2 * h =>
3.14159 * (1.37 in.)^2 * 4.5 in = 26.5 in^3
The height obtained above is an overestimate since the 'lip' of the can extends above the top of the can and the bottom of the can is indented. Thus, measure the lip and the deepest part of the indentation and subtract these measures from the height above. I measured the lip to be 1/8 inch and the indentation of the bottom of the can to be 3/8 inch. Thus, the approximation for the height is 4.5 - 1/8 - 3/8 = 4.0 inches. To get an minimum approximation of the diameter, take a piece of string and wrap it around the lip of the can. Mark and measure the length of the string. To find the radius use the formula for the circumference of the circle: C = 2 * pi * r. I measured this circumference to be 7.0 inches. Thus, the radius, r = C/2pi = 1.11 inches. Finally the volume of the can, V = pi * r^2 * h =>
3.14159 * (1.11 in.)^2 * 4.5 in = 17.4 in^3
Just what the title states, we can take the mean of the above values to approximate the solution. We must check that the answer is between the upper and lower bounds.
(22.9 + 21.7 + 21.6 + 26.5 + 17.4)/5 = 22.0 in^3
We can get a better approximation from the density of a water. Weigh a completely dry and empty soda can. Fill the can slowly with water so as not to make too many bubbles which will affect the solution. Question: how is the solution affected by the air bubbles? answer After the can is filled, wipe off any excess from the outside and reweigh. The mass of the water is the difference of these two measures. From a standard table (such as the CRC Handbook) find the density of water (which is approximately 1.0 g/mL). Then, V = m/D.
Unfortunately, I do not have access to a balance to perform this experiment.
Fill a Take an unopened soda can. with a dense material such as lead shot, copper, etc. You needn't fill it completely but just enough to get the can to sink in water. Seal the open can get a better approximation from the density of a water. Weigh a completely dry and empty soda can. Fill the can slowly with water so as not to make too many bubbles which will affect the solution. Question: how is the solution affected by the air bubbles? answer After the can is filled, wipe off any excess from the outside and reweigh. The mass of the water is the difference of these two measures. From a standard table (such as the CRC Handbook) find the density of water (which is approximately 1.0 g/mL). Then, V = m/D.
Unfortunately, I do not have access to equipment to perform this experiment.