# Cooking the perfect turkey!

With Thanksgiving coming up this Thursday, my mind has already started wondering to all the food that will be on the table.  The turkey, pumpkin pie and other festive foods seem to be beckoning me through the week.  In order to prepare for all the cooking, I thought it would be a good idea to plan ahead and see how long it will take to cook a turkey.

### Cooking

While there are plenty of recipes and different ways to prepare a turkey.  Here we will assume that we have an 8 pounds stuffed turkey.  If we cook a turkey according to the guidelines at FoodSafety.gov, this should take approximately 3 hours to reach the required temperature of 165$$^{o}$$F when cooked in a oven of 325$$^{o}$$F.  This is well and good, however, what I’d like to know is what happens if we change things up.  For example, what if you leave the turkey in too long, or take it out early?  Suppose you forgot to start it or we wanted to get it going before getting some errands done, could you adjust the temperature so that it would be done when you wanted it to be?

### Modeling

In order to answer these questions, we will first come up with a model finding the temperature of the Turkey as a function of time.  In order to do this, we will assume that the rate of change of the temperature of the turkey is proportional to the difference between the current temperature and the oven temperature.  That is, if $$T$$ is temperature, $$T_{o}$$ is the oven temperature, $$T_{i}$$ is the initial temperature of the turkey, $$t$$ is time and $$k$$ is constant, then \begin{align*} \frac{dT}{dt}=k(T_{o}-T).\end{align*}  Solving this we get that \begin{align*} \int\frac{dT}{T_{o}-T}&=\int kdt \\ \ln(T_{o}-T)&=kt+c \\ T&=T_{o}-ce^{kt}. \end{align*}

If we now use the values for the given problem we see that \begin{align*} T&=325-(325-70)e^{kt}.\end{align*}  We can use the fact that after three hours the turkey will be 165$$^{o}$$F to find that $$k=\frac{1}{3}\ln(\frac{160}{255}) \approx -.155$$.  We then have that the temperature of the turkey can be given in terms of time by the function \begin{align*}T&=325-255e^{-.155t}.\end{align*}  Graphically, this would look like

Now, in order to answer our questions regarding what the turkey would like if we left it in too long or took it out too early, we can now just plug in the time that the turkey was in the oven.  That is, if you take the turkey out after 2.5 hours, it would be 152$$^{o}$$F.  If you left the turkey in for an extra half-hour, it would be 177$$^{o}$$F.  Here, pulling it out early would be worse than leaving it long since the turkey would not have finished cooking and could potentially make you sick.  On the other hand, it may be a little over cooked, but should still be edible.  However, if you were to leave it in too long you could in fact burn it to the point it would no longer be edible.

### Changing cooking time

Now that we have determined the constant of proportionality for the rate of change of temperature for the turkey, we can now adjust the model to determine what temperature we should place the oven at in order to finish at the appropriate time.  If we want the turkey to be 165$$^{o}$$F after t hours, we can then solve for $$T_{o}$$.  Here, we would have \begin{align*} T_{o}=\frac{70e^{-.155t}-165}{e^{-.155t}-1}.\end{align*}  Graphically, we would have the following.

Therefore, if we wanted to cook the turkey in 2 hours, we should put the oven at approximately 425$$^{o}$$F.  On the other hand, if we wanted the turkey to cook over a 6 hours period, we would have to set the oven at approximately 225$$^{o}$$F.  The turkey would actually probably turn out well if you cooked it longer at lower temperatures, however, if we did turn the oven up to 425$$^{o}$$F, we would likely destroy the turkey.

The problem we face with the model is that it assumes the turkey’s temperature is the same for the entire turkey.  However, if you have ever cut a turkey too soon, you would note that the temperature between the outside and the inside of the turkey can be drastically different.  We could change the model to account for the flow of temperature through the turkey, thus giving a more accurate model.  However, I would just suggest that if you really needed to cook the turkey faster, you should cut the turkey into smaller pieces.  This would decrease the constant $$k$$, allowing the turkey to cook faster.

### Conclusion

I hope you have the best holiday season possible, and perhaps learning a bit about the way a turkey cooks will help to make your dinner a little better.  As Thanksgiving comes, I hope that you have plenty to be thankful for.

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