A Pot Of Piping Hot Stew Has Been Removed From The Stove And Left To Cool.The Relationship Between The Elapsed Time, M M M , In Minutes, Since The Stew Was Removed From The Stove, And The Temperature Of The Stew, T ( M T(m T ( M ], Measured In

Introduction

A pot of piping hot stew has been removed from the stove and left to cool. As the stew cools, its temperature decreases over time. In this article, we will explore the relationship between the elapsed time, $m$, in minutes, since the stew was removed from the stove, and the temperature of the stew, $T(m)$, measured in degrees Celsius.

Newton's Law of Cooling

Newton's Law of Cooling states that the rate of cooling of an object is directly proportional to the difference between its temperature and the temperature of its surroundings. Mathematically, this can be expressed as:

$$\frac{dT}{dm} = -k(T - T_s)$$

where $T$ is the temperature of the stew, $T_s$ is the temperature of the surroundings, and $k$ is a constant of proportionality.

Solving the Differential Equation

To solve the differential equation, we can use the method of separation of variables. We can rewrite the equation as:

$$\frac{dT}{T - T_s} = -kdm$$

Integrating both sides, we get:

$$\int \frac{dT}{T - T_s} = -k \int dm$$

Evaluating the integrals, we get:

$$\ln|T - T_s| = -km + C$$

where $C$ is a constant of integration.

Applying the Initial Condition

We know that at $m = 0$, the temperature of the stew is $T(0) = T_0$. Substituting this into the equation, we get:

$$\ln|T_0 - T_s| = C$$

Substituting this value of $C$ back into the equation, we get:

$$\ln|T - T_s| = -km + \ln|T_0 - T_s|$$

Simplifying the Equation

We can simplify the equation by exponentiating both sides:

$$|T - T_s| = e^{-km} |T_0 - T_s|$$

Since the temperature of the stew cannot be negative, we can drop the absolute value signs:

$$T - T_s = e^{-km} (T_0 - T_s)$$

Rearranging the Equation

We can rearrange the equation to solve for $T$:

$$T = T_s + e^{-km} (T_0 - T_s)$$

The Cooling Curve

The equation above describes the cooling curve of the stew. As $m$ increases, the temperature of the stew decreases exponentially.

Graphing the Cooling Curve

To graph the cooling curve, we can use a graphing calculator or software. We can choose a value for $k$ and plot the curve for different values of $m$.

Example

Suppose we want to model the cooling of a pot of stew that was removed from the stove at 100°C and left to cool in a room at 20°C. We can choose a value for $k$ and plot the cooling curve.

Choosing a Value for $k$

To choose a value for $k$, we can use experimental data or observations. For example, if we know that the temperature of the stew decreases by 10°C in 10 minutes, we can use this information to estimate the value of $k$.

Estimating the Value of $k$

We can estimate the value of $k$ by using the equation:

$$k = \frac{\ln 2}{t}$$

where $t$ is the time it takes for the temperature of the stew to decrease by 50%.

Using the Estimated Value of $k$

Once we have estimated the value of $k$, we can use it to plot the cooling curve.

Conclusion

In this article, we have explored the relationship between the elapsed time, $m$, in minutes, since the stew was removed from the stove, and the temperature of the stew, $T(m)$, measured in degrees Celsius. We have used Newton's Law of Cooling to model the cooling process and have derived an equation that describes the cooling curve. We have also discussed how to choose a value for $k$ and how to use it to plot the cooling curve.

References

• Newton, I. (1701). Opticks.
• Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids.
• Crank, J. (1975). The Mathematics of Diffusion.

Appendix

The following is a list of formulas and equations used in this article:

• Newton's Law of Cooling: $\frac{dT}{dm} = -k(T - T_s)$
• Separation of variables: $\frac{dT}{T - T_s} = -kdm$
• Integration: $\int \frac{dT}{T - T_s} = -k \int dm$
• Exponentiation: $|T - T_s| = e^{-km} |T_0 - T_s|$
• Rearranging the equation: $T = T_s + e^{-km} (T_0 - T_s)$

Glossary

• Newton's Law of Cooling: A law that describes the rate of cooling of an object.
• Separation of variables: A method of solving differential equations.
• Exponentiation: A mathematical operation that involves raising a number to a power.
• Rearranging the equation: A process of solving for a variable in an equation.
  A Pot of Piping Hot Stew: Modeling the Cooling Process - Q&A ===========================================================

Introduction

In our previous article, we explored the relationship between the elapsed time, $m$, in minutes, since the stew was removed from the stove, and the temperature of the stew, $T(m)$, measured in degrees Celsius. We used Newton's Law of Cooling to model the cooling process and derived an equation that describes the cooling curve. In this article, we will answer some frequently asked questions about the cooling process.

Q: What is Newton's Law of Cooling?

A: Newton's Law of Cooling is a law that describes the rate of cooling of an object. It states that the rate of cooling of an object is directly proportional to the difference between its temperature and the temperature of its surroundings.

Q: What is the equation for Newton's Law of Cooling?

A: The equation for Newton's Law of Cooling is:

$$\frac{dT}{dm} = -k(T - T_s)$$

where $T$ is the temperature of the stew, $T_s$ is the temperature of the surroundings, and $k$ is a constant of proportionality.

Q: How do I choose a value for $k$?

A: To choose a value for $k$, you can use experimental data or observations. For example, if you know that the temperature of the stew decreases by 10°C in 10 minutes, you can use this information to estimate the value of $k$.

Q: What is the cooling curve?

A: The cooling curve is a graph that shows the relationship between the elapsed time, $m$, in minutes, since the stew was removed from the stove, and the temperature of the stew, $T(m)$, measured in degrees Celsius.

Q: How do I plot the cooling curve?

A: To plot the cooling curve, you can use a graphing calculator or software. You can choose a value for $k$ and plot the curve for different values of $m$.

Q: What is the significance of the cooling curve?

A: The cooling curve is significant because it shows how the temperature of the stew changes over time. It can be used to predict the temperature of the stew at any given time.

Q: Can I use the cooling curve to determine the temperature of the stew at a specific time?

A: Yes, you can use the cooling curve to determine the temperature of the stew at a specific time. Simply plug in the value of $m$ and solve for $T$.

Q: What are some real-world applications of the cooling curve?

A: Some real-world applications of the cooling curve include:

• Predicting the temperature of a product during storage or transportation
• Designing cooling systems for buildings or vehicles
• Optimizing the cooling process for a specific product or application

Q: Can I use the cooling curve to model the cooling of other objects?

A: Yes, you can use the cooling curve to model the cooling of other objects. Simply replace the temperature of the stew with the temperature of the object you are modeling, and replace the time with the time you are modeling.

Conclusion

In this article, we have answered some frequently asked questions about the cooling process. We have discussed Newton's Law of Cooling, the cooling curve, and how to choose a value for $k$. We have also discussed some real-world applications of the cooling curve.

References

• Newton, I. (1701). Opticks.
• Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids.
• Crank, J. (1975). The Mathematics of Diffusion.

Appendix

The following is a list of formulas and equations used in this article:

• Newton's Law of Cooling: $\frac{dT}{dm} = -k(T - T_s)$
• Separation of variables: $\frac{dT}{T - T_s} = -kdm$
• Integration: $\int \frac{dT}{T - T_s} = -k \int dm$
• Exponentiation: $|T - T_s| = e^{-km} |T_0 - T_s|$
• Rearranging the equation: $T = T_s + e^{-km} (T_0 - T_s)$

Glossary

• Newton's Law of Cooling: A law that describes the rate of cooling of an object.
• Separation of variables: A method of solving differential equations.
• Exponentiation: A mathematical operation that involves raising a number to a power.
• Rearranging the equation: A process of solving for a variable in an equation.