In the last Geometry Problem of the Week, *Lotsa Popcorn!*, we presented the World’s Largest Popcorn Ball (as of 2006), and explained that it weighed 3,415 pounds and was 8 feet in diameter. It was claimed to be “almost 50,000 times larger than the normal popcorn balls distributed for retail consumption.”

We went on to ask students to find the diameter of a popcorn ball whose diameter is 50,000 times smaller than this, as well as the diameter of a ball whose volume is 50,000 smaller. We also asked about the weight. We received a number of solid answers to this, including these two:

**Athena B, Friends Select School**

For #s 1 and 3, I simply took the original value (big ball) and divided by 50,000. For #2, I first found the volume of the big popcorn ball, then divided by 50,000 and used the formula for the volume of a sphere in reverse to find the radius. Then I multiplied that value by 2 to get the diameter.

1. 8/0,000= 0.00016 ft, which did not seem reasonable

2. Volume of original (big ball):

V=(4/3)πr3

V=267.946667 ft3 (roughly)

volume of small:

V= 267.946667/50,000= 0.00535893 ft3

0.00535893=(4/3)πr3

r3=0.00535893/(4/3)π

r=0.108577 ft

d=0.217154 ft or 2.6″ which did seem reasonable

3. 3,415/50,000= 0.0683 lbs, which, unless my perception is off, seems reasonable enough.

**Jeremy L, Friends Select School**

1. 0.00192 inches. No, this would not be a realistic size for a normal popcorn ball. 2. 2.605848 inches. This is a realistic size for a normal popcorn ball. 3. 1.0928 ounces. This is not a realistic weight for a normal popcorn ball.

1. Multiply eight feet by twelve inches to find how many inches are in eight feet. Then, divide by 50,000.

2. V=4/3πr3

The radius is half the diameter, which would make it 4 feet.

V=4/3(3.14)(64)

V=4.1866667(64)

V=267.94667 cubic feet

Then divide by 50,000 for 0.00535893 cubic feet.

By using this new volume in the above equation, the radius can be solved for.

0.00535893=4/3(3.14)r3

0.00170667=4/3r3

0.00128=r3

0.108577=r

Since the diameter equals 2r, it is 0.217154 feet. Multiply this by 12 to get 2.605848 inches.

3. 3415 multiplied by 16 equals the number of ounces in 3415 pounds:

54,640 ounces.

This divided by 50,000 equals the answer, 1.0928 ounces.

Obviously we don’t need to give that much direction to students. We just wanted to make sure they would do something reasonable, and we figured we might need to support them in that, especially if they were working on their own without the benefit of a class or partner conversation. What we really did was make all the really interesting math decisions for the students and make it easier for us to read the solutions. We should have just presented them with the facts of the giant popcorn ball, and asked them if the 50,000 times bigger thing was reasonable. What would they compare? I’m thinking back to the So Many Salmon problem we used in Math Fundamentals earlier this year. We presented students with some facts and then asked a question, but let them determine how they would answer that question.

It makes me wonder how many people used the Scenario Only, which did only present the facts, and how many used the whole problem with the three specific questions. What would you have done? What do you think your students, whatever age they are, would have done with just the facts and the question of whether or not the 50,000 times bigger statement was reasonable?

Some ** Lotsa Popcorn!** links in case you are interested:

- The problem [requires a Math Forum PoW Membership].
- Information about accessing
*Lotsa Popcorn!*(and a selection of all our PoWs) for 21 days with a free Math Forum trial account. - Information about becoming a Math Forum Problems of the Week Member. Consider starting with a $25 membership, which gives you access to all of this year’s Current PoWs — and now you can create 36 student logins as well!