Yoplait with Areas and Volumes
by David Pleacher
Yoplait yogurt comes in two differently shaped containers.
One is a truncated cone and the other is
an elliptical cylinder (see photos below).
In this exercise, you will determine the volume of the Yoplait truncated cone in three different ways:
(1) using geometry,
(2) using the Disk Method from Calculus, and (3) using the Shell Method. You will then determine the area formula for an ellipse
using calculus, and then compute the volume of the cylinder.
Each container claims to contain six ounces of yogurt, which is equivalent to 10.83 cubic inches.
I measured the containers to the nearest tenth of an inch.
In the elliptical cylinder, the major axis was 2.5 inches, the minor axis was 2.1 inches, and it was 2.6 inches tall.
In the truncated cone, the diameter of the smaller circle (top) was 2 inches, the diameter of the larger circle (bottom) was 2.4 inches, and it
was 2.8 inches tall. In the latter, the height is deceiving because the container is recessed on the bottom; the 2.8 inches is the height of
the portion of the container that holds the yogurt - the actual container is over three inches tall.
(1) The Yoplait Truncated Cone Container - Using geometry to find its volume
First extend the sides of the container until it forms a cone, as pictured in the diagram below.
Let h = the height of the truncated cone.
Let x = the height of the new cone that is formed by extending the sides.
Solve for x.
Now solve for the volume of the truncated cone by subtracting the volumes of the two
cones.
How does this answer compare to 10.83 cubic inches (the six ounces of yogurt)?
(2) The Yoplait Truncated Cone Container - Using the Disk Method to find its volume
In the diagram in (1), let R represent the positive x-axis, and h + x represent the positive
y-axis.
Let the intersection of those two segments represent the origin.
Now solve for the equation of the line representing the lateral edge of the cone.
Now determine the volume obtained when rotating that line about the y-axis from y = 0
to y = 2.8".
How does this answer compare to the answer obtained in (1)?
(3) The Yoplait Truncated Cone Container - Using the Shell Method to find its volume
Now determine the volume of the truncated cone by using the Shell Method.
Note that this must be done in two steps. Find the volume from x = 0 to x = 1,
and then find the volume from x = 1 to x = 1.2". The height of the cylindrical shells from
x = 0 to x = 1 is a constant, but the heights diminish from x = 1 to x = 1.2.
How does this answer compare to the answers obtained in (1) and (2)?
(4) The Yoplait Elliptical Cylinder Container -- Find the area of an ellipse
Use calculus to determine the area of an ellipse with major axis 2a and minor axes 2b.
Use the formula 
as the formula for the ellipse, and integrate the function in the first quadrant from x = 0 to x = a, and then multiply by four.
You will need to use the method of Trigonometric Substitution and some trig identities.
(5) The Yoplait Elliptical Cylinder Container -- Find the volume
Now, using the formula obtained in (4), multiply by the height to get the volume of the
cylinder.
How does this answer compare to the answers obtained in (1), (2), and (3)?