Many of you are probably on your way back home for the holiday weekend. I will also be spending time with my family. However, just because there is a holiday I don’t stop doing mathematics. And neither should you!
Last weekend I was looking a turkeys at Safeway and I became fascinated by the labels. I posted this analysis of Butterball Turkey cooking times in my College Algebra blog to encourage students to look at everyday objects in a mathematical perspective.
Every turkey in the meat section included a table of weights and corresponding cooking times. It seemed that each brand of turkey had slightly different cooking times. Shouldn’t the tables all be the same? After all. turkey is turkey.
So I began to look online at different tables for cooking a fresh turkey. The table below is one I found on the Butterball website.
Let’s make a scatter plot of the unstuffed cook time as a function of the weight. Examining the first two columns, we see that the weights and cook times are given as a spread of values. However, note that the end of the spread of weights in the first row is 7 pounds and the start of the next row is 7 pounds. The weight of 7 pounds corresponds to a cooking time of 2 1/2 hours. Examining the other rows allows us to form the table below.
|Weight (lbs)||Cook Time (hours)|
Now graph these data with the weight as the independent variable and the cook time the dependent variable.
As you might expect, the slope of the line obtained by linear regression is positive. This indicates that as the weight increases, the cooking time increases. The exact relationship between increases in weight and increases in cook time is expressed in the slope.
Now let’s look at the cook time for stuffed turkeys. If we try to create a table of weights and cook times from the first and third columns of the table above, you notice an inconsistency.
In this table, a 9 pound turkey corresponds to TWO different cook times! In the second row, the cook time for a 9 pound turkey is 4 1/2 hours and in the third row the cook time for a 9 pound turkey is 3 3/4 time. Which cook time should we use? If you were to pick the wrong time, could you potentially end up with a 9 pound piece of turkey leather?
To come up with the correct time for a 9 pound turkey, let’s create another scatter plot. We’ll leave out the cook time for a 9 pound turkey and use the linear trend to interpolate the correct cook time.
The data for stuffed turkeys lie very close to the line y = 0.1572x + 1.6111 where x is the weight in pounds and y is the cook time in hours. We can use this function to estimate the cook time for a 9 pound turkey,
y ≈ 0.1572(9) + 1.6111 ≈ 3.0259
This cook time corresponds to 3 hours and 1.5 minutes. Since the times in the table are given to the nearest quarter hour, the time for a 9 pound stuffed turkey should be 3 hours.