Time to roast a turkey

2021-11-25

I saw on mathematics Twitter a claim that if you take the temperature of a turkey in the oven twice, you can predict when the turkey will reach the correct temperature to be fully roasted. This was based on Newton's Law of Warming, which says the rate the temperature increases is proportional to the difference in temperature between the turkey and the oven. They said that they tried this, and it predicted their turkey would be done at 6:30pm, and the prediction was spot on.

I worked through the math myself, and came up with the formula for the time it will take the turkey to reach the correct temperature.

Here is the formula:

• Let $T_{\text{oven}}$ be the oven temperature.
• Let $T_{\text{done}}$ be the desired temperature of the turkey when it is done.
• Let $T_1$ be the temperature of the turkey, when measured at a certain time.
• Let $T_2$ be the temperature when measured $t$ hours after the first temperature.

Then the amount of time until the turkey is done (after the second temperature is taken) will be
$$\text{time}=\frac{At}{B}-t,$$
where
$$A=\ln \left|\frac{T_{\text{oven}}-T_{\text{done}}}{T_{\text{oven}}-T_1}\right|$$
and
$$B=\ln \left|\frac{T_{\text{oven}}-T_2}{T_{\text{oven}}-T_1}\right|.$$

For example, suppose the oven temperature is $T_{\text{oven}}={350}^{\circ }$ and the desired temperature is $T_{\text{done}}={180}^{\circ }$. Suppose at a certain time, the temperature of the turkey is $T_1={71.6}^{\circ }$, and $t=2$ hours later, the temperature is $T_2={132.4}^{\circ }$. Then we have
$$A=\ln \left|\frac{T_{\text{oven}}-T_{\text{done}}}{T_{\text{oven}}-T_1}\right|=\ln \left|\frac{350-180}{350-71.6}\right|=\ln 0.6106=-0.4932,$$
$$B=\ln \left|\frac{T_{\text{oven}}-T_2}{T_{\text{oven}}-T_1}\right]=\ln \left|\frac{350-132.4}{350-71.6}\right|=\ln 0.7816=-0.2464$$
so
$$t=\frac{At}{B}-t=\frac{2(-0.4932)}{-0.2464}-2=2.004$$
hours. So the turkey will be the correct temperature $2$ hours after the second temperature was taken. Notice we did not need to know the initial temperature of the turkey, and we did not need to know how long after the time the turkey was put into the oven the first temperature was taken.

By the way, this example is built around the following test formula, which gives the temperature $T$ of the turkey $t$ hours after it is put into the oven:
$$T=350-315e^{-0.12335t}$$
This formula is typical for solutions of problems based on Newton's Law of Warming. This formula is rigged to have the temperature increase from ${35}^{\circ }$ to ${180}^{\circ }$ after $t=5$ hours, and it gives ${71.6}^{\circ }$ at $t=1$ hour, and ${132.4}^{\circ }$ at $t=3$ hours. The formula aboves gives the correct time it took until it reached ${180}^{\circ }$, which reassured me that the formula is correct.