Question

b - 16: eeeeerw! hannah’s volleyball team left their egg salad sandwiches sitting in their lockers over the weekend. when they got back on monday they were moldy. “perfect!” said hannah. “i can use these sandwiches for my biology project. i’ll study how quickly mold grows.” using a transparent grid, hannah estimated that about 12% of the surface of one sandwich had mold on it. she threw the sandwich out. for the rest of the week, hannah came back when she had time. each time she measured somebody else’s sandwich and threw it out. she collected the following data:

day 1 (monday)	day 2 (tuesday)	day 2 (tuesday)	day 4 (thursday)	day 4 (thursday)	day 4 (thursday)	day 5 (friday)

a. create a scatterplot and skitch it. is a linear model reasonable?

b. based on the story, what kind of equation do you think will best fit the situation?

Explanation:

Step1: Prepare data for scatter - plot

Let the day number be the independent variable $x$ (where Day 1 is $x = 1$, Day 2 is $x=2$, etc.) and the percentage of mold be the dependent variable $y$. The data points are $(1,12)$, $(2,15)$, $(2,13)$, $(4,26)$, $(4,27)$, $(4,24)$, $(5,38)$.

Step2: Analyze linear - model suitability

When we plot these points on a scatter - plot, we can observe that the points do not seem to lie approximately on a straight line. There are multiple points for the same $x$ - value (e.g., for $x = 2$ and $x = 4$) with different $y$ - values, and the overall pattern does not show a clear linear trend. So a linear model is not reasonable.

Step3: Determine best - fit equation

In the context of mold growth, mold typically grows exponentially over time. As time passes, the mold has more time to multiply, and its growth rate is proportional to the amount of mold present at any given time. So an exponential equation of the form $y = ab^{x}$ (where $a$ and $b$ are constants, $a>0$, $b > 1$) would best fit the situation.

Answer:

a. When creating a scatter - plot of the data points $(1,12)$, $(2,15)$, $(2,13)$, $(4,26)$, $(4,27)$, $(4,24)$, $(5,38)$, a linear model is not reasonable as the points do not form a linear pattern.
b. An exponential equation of the form $y=ab^{x}$ ($a>0$, $b > 1$) would best fit the situation.