Subject. How to prepare a medium rare roast, mathematically.
Please complete the exercise first before moving forward.
Exercise. Given a 1.82-kg roast, initially at 283K, is placed in a 464K oven at 17:00. After 4500 seconds it is found that the temperature T(t) of the roast is 325K. When will the roast be 339K (medium rare)? [Source: Edwards & Penney. Elementary Differential Equations with Boundary Value Problems. NJ: Pearson. 5th ed, 2004. Pg 39.]
Note: I have not utilized MLA format in a very long time, I forgot the order and probably omitted certain information. However, who cares? This is not a research paper.
Solution. Using Newton's Law of Cooling, we have dT/dt = k(464 - T). Solving the differential equation, we get t = 6300 (sec).
A set of objects required:
{
- Pencil and paper (may be omitted for the especially-enlightened),
- Newton's Law of Cooling (or mastery of thermodynamics),
- Internal kinetic energy measurement instrument (reads in Kelvins),
- One (1) heating unit (settings in Kelvins),
- One (1) chronometer (reads in seconds),
- One (1) balance (reads in kilograms),
- Mastery of differential equations (or an extremely bright chauffeur),
- A roast.
}
General solution. A ordered 9-tuple of steps needed to be taken:
(
- Measure the temperature and mass of the roast and write it on the paper with a pencil,
- Put the roast in the heating unit and tune its temperature setting to above the critical level (339K for medium rare). The bigger the difference betwix the seeting and the critical level, the shorter the interval of which the roast must remain inside the space closed by the heating unit,
- Take note of the temporal coordinate as indicated on the chronometer and record the figure on paper,
- After a finite time interval, big enough to reduce error and small enough to prevent over-cooking, use the chronometer on the roast and take note of the temperature reading,
- Take note of the chronometer again (and record on paper),
- Write a differential equation by Newton's Law of Cooling and the data collected,
- Solve the differential equation (with or without the help of thy chauffeur),
- Take note of the chronometer and compare: if the reading from the chronometer is larger than or equal to the calculated value, eject(1) the roast and poweroff(8) the heating unit; otherwise,
- Sleep(1) for one (1) second and repeat last step.
)
Footnote. Since this was a HOWTO-type document, I did not weave in the motivation of the steps. Many should be clear, e.g. recording the data on paper for fear of loss of information due to faulty human RAM. One thing deserves further explanation. Taking the temperature of the roast after a finite time elapsed provides information to calculate the cooling constant k in Newton's Law of Cooling. If one performs this experiment many times, assuming identical pieces of roast, he may skip this step and utilize previous results of k. However, it shall be noted that, even with laziness-driven human technological development, it is still a daunting task to produce identical lab samples and equipments, especially when organic tissue is involved.