Using extrapolation, measurements from graduated cylinders every 3 minutes, and clocks, we concluded that “Frosty” was put to death about 52 minutes before 10:17 when we started observing the rate at which “his” ice melted, which was about 9:25 a.m. The rate at which “he” melted approximated the line y= (9/22)x + 21+(9/22).
Next, we played with skittles, representing radioactive atoms, shook 80 around in a cup, and separated out the ones with the little “s” down, which represented atoms that decayed, until there were no more left.
This type of graph is an exponential decay function and all half-lives can be modeled using them. The half-life of the skittles, from my group’s experiment, is about ten seconds (the time for us to shake a cup and pour the skittles onto a napkin for counting, record the number with the “s” up, and put those skittles back into the cup to start again). There is no good linear relationship between the number of cup tosses and the “radioactive nuclei”, but the negative logarithm of the number of remaining radioactive nuclei (on the right in the picture above) gave a line approximating y= .25x- 2. The only linear thing about this graph is the ratio by which it decreases each turn, which is about 1/2.
One similarity between this experiment and the “Frosty” experiment is that the amount of radioactive nuclei and the amount of ice strictly decreased. A second is that the rate and ratio of decay were not 100% linear, but were faster between their surrounding points sometimes and slower other times.
One difference between the two is that “Frosty’s” follows a line much more than the skittles graph. A second is that the amount of ice decreased by a near-constant amount per interval, while the skittles decreased by a near-constant ratio per interval; one is subtraction while the other is division.