Mathematics: Roast a turkey, need how long arh?

Okay, I confess that I never roast a turkey or eat a turkey before...but a chicken, yes!

It just so happens that when I'm was studying this book: First Course in Mathematical Modeling by Frank R. Giordano...I came across this interesting example.

This book in page 324 stated that

One of the general rule for roasting a turkey is the following: Set the oven to 400 ° F (204.44 °C) and allow 20 min per pound for cooking. 

1 pound = 0.4536 kg , 1 kg = 2.2462 pound

How good is this rule?

So, this book started with some assumptions, model the formula and testing the results.

Please be reminded that assumptions are not always hold, formula is not always true and the results are probably not always close to what we want but we are trying to approximate the results here...

Let:

t = cooking time for the turkey

*t certainly depend on the size of the turkey (all the turkeys certainly have different size) but we assume turkets are similar.

l = length of the turkey ( uncooked meat)

ΔT_m = the difference between the temperature of the raw meat and the oven

ΔT_c = the difference between the temperature of the cooked meat and the oven

k = coefficient of heat conduction for a particular uncooked food

Model:

t = f(ΔT_{m ,} ΔT_c, k, l)

So, after some extensively using some tricks we arrive at the formula

where t ∝ W^2/3

I'm not going to derive the whole formula here because I'm sure nobody going to bother about it. The W suddenly appear in the formula because we know W ∝ l³ . 

*The main reason is I'm lazy to type the whole thing out*

 

Just convince yourself the formula is like that. =)

 This formula t ∝ W^2/3 means the required cooking time is proportional to weight raised to the two-thirds power.

 Be patient with me, I almost done...

Take for an example:

If  
t₁ hours are required to a turkey weighing W₁ pounds and 
t₂ = is the time for a weight of W₂ pounds, 
then our formula become...

 

Let's consider we finally cooking a  real 10 kg turkey vs a real 3 kg chicken. 

According to the our rule, the ratio of cooking time is given by

 

But, if we use our formula, after we substitute into it

So our rule tells us that it will take 3.33 times as long to cook a 10 kg turkey compare to cook a 3 kg chicken and our formula tells us it will only take 2.23.

Which one is correct?

I don't know. We certainly need to test our formula or our model to make sure it is correct. Testing the results part is in the book, please read if you want to know but I'm not really sure whether our book gives a correct from a real world data or not.

I just want to say...finally we have at least a formula for cooking a turkey!