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:
Then the amount of time until the turkey is done (after the second temperature is taken) will be
where
and
For example, suppose the oven temperature is
so
hours. So the turkey will be the correct temperature
By the way, this example is built around the following test formula, which gives the temperature
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