Math in the News

Thanksgiving Temps

What You Will Learn

• Summary– In this issue of Math in the News we look

at graphs representing the thawing of a frozen Thanksgiving turkey.

• Define graphs of cooling curves• Explore graphs of warming curves• Model data using logarithmic

functions

Cooling Curves

• You’re probably more familiar with cooling curves, which can be modeled by decreasing exponential functions.

Thawing Curves

• The situation with a frozen turkey is different. You start with a temperature below freezing and want to reach a higher temperature.

Mathematical Modeling

What mathematical model can be used to show the thawing of a turkey?

Discuss as a class how you would model this temperature increase.

Thawing Curves

• It turns out that a reasonable model for the reverse of cooling is the inverse of the exponential, an increasing logarithmic function.

Thawing Curves

• The USDA suggests two methods for thawing a turkey:– Thawing it in a refrigerator– Thawing it in a container of water

• You must AVOID thawing a turkey at room temperature on a counter. The reason is that bacteria can form on the thawed parts long before the rest of the turkey has thawed.

Guideline for Thawing a Turkey

Refrigerator• Whole turkey:

– 4 to 12 pounds — 1 to 3 days

– 12 to 16 pounds — 3 to 4 days

– 16 to 20 pounds — 4 to 5 days

– 20 to 24 pounds —5 to 6 days

Container of Water• Whole turkey:

– 4 to 12 pounds — 2 to 6 hours

– 12 to 16 pounds — 6 to 8 hours

– 16 to 20 pounds — 8 to 10 hours

– 20 to 24 pounds — 10 to 12 hours

Thawing Curves

• We’ll use the USDA guidelines to build a logarithmic function to model the thawing of a frozen turkey.

• What you’ll need: – A graphing calculator, preferably one that

has variable sliders– Thermometer– Spreadsheet for gathering data

Thawing Curves: Step 1

• Pick a method for thawing. Identify the method your family will be using to thaw the Thanksgiving turkey.

• For this example, let’s use the refrigerator method to thaw a 12-pound turkey over 3 days. Use these data points as the start and end points for the data:

(0, -10) and (3, 35)At time 0, the temperature is -10°F.

After 3 days the temperature is 35°F.

Thawing Curves: Step 2

• Graph the start and end points on your graphing calculator.

Thawing Curves: Step 3

• On your graphing calculator, set up a logarithmic function of the form:

• Make sure that the values for a, b, and c are tied to slider values on your graphing calculator. With sliders you can try endless values for a, b, and c and see the impact on the graph.

Thawing Curves: Step 4

• Manipulate the sliders until you find a logarithmic function that fits the end points. (There are an endless number possible.)

Thawing Curves: Step 5

• Use the Trace feature to generate a table of values. Use these to compare predicted values from the model to measured values.