# Popcorn Math

Everyone loves popcorn. People have been popping corn for 5000 years. Over the last century, there has been an expanding market for mass produced popcorn in the United States and around the world. There are multiple ways to pop popcorn. The original way would be to cook the kernels over a heat source such as a fire. This developed into cooking corn over a stove and eventually evolved into mass producing product such has Jiffy Pop. With the development of the microwave, making popcorn went from being a clumsy time consuming process, to clicking a button. But….

How do you know how long to click that button for?

To answer this question, we designed an experiment using a standard recipe for making your own popcorn at home. All you do is put kernels of popcorn in a paper bag, fold the top of the bag over, and put it in the microwave. The question we were looking to answer was “How long do you put the bag of popcorn in the microwave for?”. At first, we tried this experiment with a 1/4 cup of kernels in every bag. This didn’t quite work out as planned.

Original Trial and Errors

We were trying to make each time as realistic as possible by using an online recipe for homemade popcorn, which consisted of simple ¼ cup of plain kernels in a paper bag. We assumed that ¼ cup would have the same amount of kernels in each cup, which we counted to be 380. However, this was not true as we found in one specific trial upwards of 450 kernels. Because of inaccurate measurements and differences in kernel size, the real number varied from trial to trial. This cause the data to fluctuate in a non logistic pattern, so we decided we needed to start over.

Effects on Optimal Time

We determined that there are several factors which affect the amount of time it takes for the kernels to pop and burn. The number of kernels popped at one time is the main cause. We first tested ¼ cup where we found all popcorn was completely burnt at 4 minutes and ten seconds. We then changed the amount to 100 kernels per bag, but at 4 minutes 15 seconds only 7 kernels were burned in our first trial and only 2 in our second. Even in our largest trial, 5 minutes, only 11 of the 100 burned in the first trial and 9 in the second. We can conclude from this data that the fewer kernels there are in a bag, the longer you will need to set your timer.

Another factor that affects each individual kernel is the moisture content of the kernel. The amount of water in each kernel determines how long each kernel will take to pop because the microwave vaporizes the water causing the kernel to expand. If there is more water in a kernel it will take longer it to pop. However, the lower the water content, the less likely the kernel is to pop. While you can calculate the moisture content, you can’t control it. The only way we would be able to calculate it with the technology available to us would be to weigh each kernel before and after popping, subtracting the difference and finding the percentage of water. But because we still only would know the content after the trial is complete it is not a effect control. Therefore, time and technology forced us to ignore this variable and assure it was constant.

Using the function $P(t) = \frac{K P_0 e^{rt}}{K + P_0 \left( e^{rt} - 1\right)}$   we graphed our data, and found the function that models our data the best. $P(t) = \frac{K P_0 e^{rt}}{K + P_0 \left( e^{rt} - 1\right)}$ has variables; K-constant, Po-Initial popped kernels, r-rate, t-time (in seconds)
Although there are no popped kernels when we start our trial, we cannot use zero for Po because all values on the line are zero when we do so. Because we could not use zero, we tried using other numbers close to zero but found that when we used .01 for Po, the line of best fit had the highest correlation value.
For the first trial, the line fit our graph pretty well and had a correlation value of R^2=.904 K= 40.922     r=.048932
In the second trial, the line fit our graph the best out of any trials. We had a correlation value of R^2=.994 K=43.342     r=.04891
In our third and final trial, the again fit pretty well, but not quite as well as the second trial. We had a correlation value of R^2=.988 K=44.202     r=.047704
All of our data on the same graph shows that our trials were very similar, and for the most part the best fit lines were alike. Even though while recording it didn’t seem it, our data was fairly accurate. After looking at all of our data, and of course taste testing all of the popcorn to see which trials were really the best, we came to a conclusion. Although we never reached a point were burning the popcorn was a worry, the popcorn did eventually get crispier than preferred. The best tasting popcorn and most popped kernels occurred around three minutes. Although we did have a couple crispy pieces of popcorn, three minutes is the optimal time to pop your popcorn.

Desmos- https://www.desmos.com/calculator/sv9rggmkop