Activity: Geometric Shapes, Area and Weight of M&Ms Candy

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Activity Title: How Many Are In The Jar?

Topic: Geometric Shapes, Area, Volume, and Weight

Objective: Creating the proper engineering views to describe an object.  

Material: M&M’s

Introduction: An oblate spheroid is a surface of revolution obtained by rotating an ellipse about its minor axis.

three candy

An oblate spheroid is a three dimensional solid and is most easily described as a sphere that has been compressed from top to bottom, causing the equator to bulge out. The Earth, and most rotating bodies in space are shaped like this. Forces that act on the Earth as a result of its rotation produce this shape. These forces cause the Earth’s mass to try to fly outward as the planet rotates, but gravity holds it together.

An ellipse rotated around its minor axis will describe the three dimensional object known as an oblate spheroid. sphere 2The ellipse is an oval-shaped two-dimensional construct, defined in geometry as the shape that results from the intersection of a flat plane with a cone. It has two axes: major and minor. A line passing through the center of the ellipse and having its end points positioned the maximum distance apart is the ellipse’s major axis and is its maximum possible diameter. The minor axis passes through the center of the ellipse, has endpoints positioned the minimum distance apart, and is the minimum possible diameter of the ellipse.

The earth’s shape is a spheroid

Although the earth’s shape is technically an ellipsoid, its major and minor axes do not vary greatly. In fact, its shape is so close to a sphere that it is often called a spheroid rather than an ellipsoid.

spheriod

A spheroid is simply an ellipsoid that approximates a sphere. These examples are two common world spheroids used today with their values rounded to the nearest meter. For each spheroid, the difference between its major axis and its minor axis is less than 0.34 percent.

  1. An ellipsoid that approximates the shape of a sphere
  2. An ellipsoid created by rotating an ellipse about either its major axis (called a prolate spheroid) or its minor axis (called an oblate spheroid)

STEP 1:

Procedure: Use the provided “M&M’s” to sketch the CANDY AS YOU SEE IT RIGHT OUT OF THE Package front, top, right side, and isometric views in the space provided.one M&M

ISO_paper2

Questions:

  1. Explain the problem with the views? (Hint: hidden lines)
  2. Is it necessary to draw the three primary views? (top,front,side)
  3. If you were to draw the other candy pieces, will the views be the same?
  4. Why is the front view the side with the letter M?

 

 

STEP 2:

Procedure: Use the provided “M&M’s” and cut it in half directly through the center. Now sketch the CANDY AS YOU SEE it after it has been cut in half front, top, right side, and isometric views in the one M&Mspace provided.

ISO_paper2

QUESTIONS:

  1. Explain why section views are important.
  2. Explain what type of section view this is.
  3. What do the arrows on the cutting plane line represent?
  4. Why do we change the angle of the section lines when there are multiple materials in an object?

 

STEP 3:

Procedure: Calculate the Weight of an M&M’s

Spheroids: If any two of the three axes of an ellipsoid are equal, the figure becomes a spheroid (ellipsoid of revolution).

There are two kinds of spheroid: 1. oblate spheroid (M&M’s) and 2. prolate spheroid (Football)

oblate_prolate

 

 

 

 

Calculating the Volume of a spheroid:

Weight of sphere

Given: What is the volume of a spheroid with a major axis (a) of 2” and a minor axis (b) of 1”?

(Show Work) ______________________________________________________________

Using your measurements from the dimension sketch above, calculate the volume of your M&M:

(Show Work) ______________________________________________________________

 

Given: The density of an M&M is 1.7 grams per cubic centimeter. Not all M&M’s have the exact same density, but they are all pretty close. (Wd=V*D)

What is the Weight of an M&M =

(Show Work) ______________________________________________________________

If a package of M&M’s weigh 0.45lbs, how many M&M’s are in the package? (Ignore the weight of the package itself.

(Show Work) ______________________________________________________________

STEP 4: How many M&M’s are in the jar?

Have you have wondered how many standard sized M&M’s would fit in a container? Perhaps you had jarthe luxury of participating in a contest during your youth where the object was to guess the number of M&M’s in a container. How did you fair? What approach did you take to figure it out? Well, there are many approaches to solving this problem. However, the most accurate method will likely involve the use of a formula derived from measurements of M&M’s and their packing ratio (the percentage of space M&M’s take up in a container)

Given: American manufacturing processes are a marvel of modern engineering. Even so, there will always be some variability between every item that is produced on an assembly line. In the case of M&M’s, the size, shape, and mass of each candy will likely vary to some degree. However, the Mars Incorporated has strict quality control standards and that the variability of these characteristics will likely be very small. With that said, the average value for an M&M’s mass, volume, and packing ratio (this relates to the shape) are. The results are below:

average m&M

 

Formula For Computing Quantity of M&M’s Based on a Container’s Volume

Using the average values shown above, the following formula can be derive from the information to calculate the number of M&M’s in a container based on its volume.

mmimage

QUESTIONS: How many M&M’s are in a 250 milliliter jar?

(Show Work) ______________________________________________________________