However, as I left the theater, I noticed that I didn't actually eat all of the popcorn. There was about two and a half inches of popcorn left at the bottom of the bucket. I could take it home with me, but stale popcorn doesn't sound too appetizing and I decide to throw it away. Did I just get ripped off? Should I have just bought the small popcorn?
There's a couple of ways of modifying this task to address the needs of different grade levels. It all depends on what information is given to the students. If you can just give the students the number of cups of popcorn in each bucket, then this is a fairly simple unit price problem. If you just give dimensions of the buckets, you will need to derive and use formulas. It would also be extremely helpful to use a spreadsheet.
6th Grade Version:
Info required...
Questions to explore...
What is the unit price for each size?
What is the percent change in size, price, unit price?
What is the least amount of popcorn from the large container (in cups) you would need to eat so that you don't get ripped off? (This is not as interesting a question as the 8th grade version because you can't usually tell how many cups of popcorn are left in a bucket.)
8th Grade (or beyond) Version:
Info required...
Volume of a truncated cone:
You will notice that there is a little bit of popcorn above the rim of each bucket. There is also a small gap on the bottom of each bucket. I assumed that the added and subtracted volumes of this popcorn would more or less cancel each other out. I could be wrong about this!!!
Questions to explore...
What is the capacity of each size?
What is the unit price for each size?
What is the percent change in size, price, unit price?
How many inches of popcorn would be left in the large bucket if you eat just as much as the small bucket?
What is the least amount of popcorn from the large container you would need to eat so that you don't get ripped off? In other words, how many inches of popcorn can I leave at the bottom of the bucket?
The answer....
I'm not leaving my full solution here because I'm curious to see how others might solve it. Basically, I used a spreadsheet to test different heights of popcorn eaten to determine where the unit price of the large matches the unit price of the small. If you think about it, this is further complicated because as you eat popcorn, the height AND top radius changes. You will have to come up with a formula that calculates the top radius based on the height.
I determined that you get ripped off if you leave more than two inches of popcorn at the bottom of the bucket.
