Kristen Robinson

EMAT 4680

7 November 2000

Lab Write-up 3

Title:  Volumes of Cones

Team Members:  Megan and Lee

Materials:  worksheet with pre-drawn circles for cones, scissors, tape, protractor, ruler, regular             size bag of m&m’s

Procedure:

            1.  Cut out each of the circles from the worksheet.

            2.  Measure the radius of each circle (in centimeters).  Record each value on a separate                                   line of the independent variable column.

            3.  Cut a slice in each circle from the rim to the center of the circle.  Repeat steps 4 - 7 for              each circle.

            4.  Mark an arc of 135 degrees from one edge of the slice.

            5.  Fold the second edge of the slice to the 135 degree mark.

            6.  Tape the edges together.  The result should be a cone.

7.  Fill the cone with M&M’s until the cone is reasonably full.  This may require

skimming off the top of the M&M’s.  Record the number of M&M’s used in the dependent variable column.

Data Set:

Independent Variable

Dependent Variable

Radius (cm)

Volume (# M&M's)

2

3

3

9

4

21

5

39

 

Scatter Plot:

Analysis of Data:

The best function to model the experimental data is y = 3x^2 - 9x + 9.  This is a quadratic polynomial.  The function can also be written as y = x^2 - 3x + 3.  The formula for the volume of a cone is V = (1/3)(pi)(r^2)h.  Because the height of the cone changes constantly as the radius changes (the cut arc was always 135 degrees), the only variable in the volume function is r.  Therefore, a quadratic polynomial accurately models the data.  Even creating a scatter plot on Microsoft Excel and adding a trendline which is a polynomial of degree 3, the obvious choice, results in the same equation as above.  The domain of the above function should be limited to

x > 0 because a negative radius could never exist.  The radius also could not equal 0 because then the circle would not exist.  By limiting the domain as such, the range is automatically limited to

y > 3.  By finding and solving the derivative, y’ where y’ = 0, the vertex of the quadratic graph is found to be at (3/2, 3/4).  Therefore, the radius of 3/2 cm yields the smallest volume of 3/4 of an M&M.

Conclusions & Extensions/Predictions:

Errors could occur in the experiment.  Air pockets will exist between the M&M’s, so it is impossible to find the exact number of M&M’s used to completely fill the cone.  There is not a uniform number of air pockets in each cone, so the error could affect results.  It also is impossible to place the M&M’s inside the cone such that the M&M’s are completely level with the top of the cone.  This results in another error in the exact amount of M&M’s necessary to completely fill the cone.  This exercise is nice for students, though, because when they finish the lab they have M&M’s to snack on.