Team Members
I received this data set from Elizabeth Jones, Raju Patel, and Corrie Collier.
Materials
This lab required M&M candies, paper, tape, scissors, a compass, a ruler, and a protractor.
Procedure
Draw and cut out paper circles with different radii. (We used circles with integer valued radii from 2 to 5 cm). Measure, cut out and remove a 135 degree sector from each circle. Use the remaining portion of each circle to form a cone. Measure the volumes of the cones by counting the number of M&M candies need to fill the cone.
Data Set
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Scatter Plot
Analysis of Data
Let x equal the measure in centimeters of the radius of the original circle from which the cone was formed. Let y = f(x) equal the volume of cone measured by the number of M&M candies necessary to fill the cone. The domain of the function can be any nonnegative rational number, but integer values from 1 to 20 are most practical. The range of the function is contained in the nonnegative integers since we are only using whole M&M's.
At first glance, the scatter plot of our data does not appear to differ a great deal from the previous exponential functions we have analayzed. However, our experience with volume of a figure as a function of the meausure of one dimension of the figure suggests that we have a cubic function.
We use the cubic regression curve with y-intercept set at 0 (since 0 radius of circle means 0 M&Ms inside the cone).
Visual constrast of the fit as well as the strength of the correlation factor reveal that a cubic function is indeed a much better fit than than an exponential.
Since my memory of volume formulas motivated my choice of a cubic functions, I should be able to more explicitly algebraically relate the behavior of our data to how we calculate the volume of a cone.
The cones formed in this lab appeared to be right circular cones whose volume can be calculated by the formula Volume = V = (1/3)1R^2 H where R is the radius of the cone (different from the radius of the circle , x) and H is the perpendicular height from the point to the center of the base. The length of the side of the cone is x ( the radius of the original circle). But since the cone is right, x^2 = R^2 + H^2.
The circumference C of the base of the cone is (360 -135)/360 =5/8 of the circumference of the original circle, (5/8)21x. But the C = 21R , so R=5/8x.
So H = square root [x^2 - R^2]
= square root [x^2 - (25/64)x^2]
= square root [(39/64)x^2]
= (square root [39]/8)x.
Therefore V = (1/3) 1 ((5/8)^2)(x^2) (root[39]/8)x = .319323565 * x^3.
Let's compare this function with our data. Since our x values are measured
in cm, when we input them into our algebraic function, we will get volume
in cubic cm, not M&M's. However, we find the ratios of V(x)/f(x) consistently
close to 1.2. That means that the volume of an M&M is roughly 1.2 cubic
cm.
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So, we have derived an analytic formula that closely resembles the behavior of our data set!
Volume of Cones in M&Ms = VM(x) = 1.2*.319323565 * x^3.
Conclusion
In light of our previous experience with volume formula that involves cubing one dimension of a figure or multiplying together three dimensions together, it is not all surprising to see that our volume data represented a cubic function. It is interesting to note the expected relationship holds even when measuring a dimension in the traditional "ruler" unit of centimeters but measuring volume in the very unconventional unit of M&Ms. The only necessary consideration is multiplication by a simple conversion factor between traditional cubic centimeters and creative M&Ms. Perhaps this observation, which probably shouldn't have been so surprising to me, reveals that volume is intrinsically a cubic function of the length of a dimension, not just a result of the units and formulas that we often use to discuss volume and length.