The Table Represents The Temperature Of A Cup Of Coffee Over Time.$[ \begin{array}{|c|c|} \hline \text{Time (minutes)} & \text{Temperature (degrees Fahrenheit)} \ \hline 0 & 200 \ \hline 10 & 180 \ \hline 20 & 163 \ \hline 30 & 146
Introduction
The table below represents the temperature of a cup of coffee over time. In this analysis, we will explore the mathematical concepts that govern the cooling of the coffee and provide insights into the underlying physics.
The Table
| Time (minutes) | Temperature (degrees Fahrenheit) |
|---|---|
| 0 | 200 |
| 10 | 180 |
| 20 | 163 |
| 30 | 146 |
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:
dT/dt = -k(T - Ts)
where:
- T is the temperature of the object at time t
- Ts is the temperature of the surroundings
- k is the cooling constant
Applying Newton's Law of Cooling to the Coffee
We can apply Newton's Law of Cooling to the coffee by assuming that the temperature of the surroundings is constant at 70°F (20°C). We can then use the data from the table to estimate the cooling constant k.
Let's assume that the temperature of the coffee at time t is given by the equation:
T(t) = Ts + (T0 - Ts)e^(-kt)
where:
- T0 is the initial temperature of the coffee (200°F)
- Ts is the temperature of the surroundings (70°F)
We can use the data from the table to estimate the value of k. For example, at time t = 10 minutes, the temperature of the coffee is 180°F. We can substitute this value into the equation above to get:
180 = 70 + (200 - 70)e^(-k(10))
Simplifying the equation, we get:
110 = 130e^(-10k)
Dividing both sides by 130, we get:
0.846 = e^(-10k)
Taking the natural logarithm of both sides, we get:
-0.173 = -10k
Dividing both sides by -10, we get:
k = 0.0173
Modeling the Cooling of the Coffee
Now that we have estimated the value of k, we can use the equation above to model the cooling of the coffee over time. We can substitute the value of k into the equation to get:
T(t) = 70 + (200 - 70)e^(-0.0173t)
We can use this equation to predict the temperature of the coffee at any time t.
Graphing the Cooling of the Coffee
To visualize the cooling of the coffee, we can graph the equation above. We can use a graphing calculator or software to plot the temperature of the coffee over time.
Conclusion
In this analysis, we have used Newton's Law of Cooling to model the cooling of a cup of coffee over time. We have estimated the value of the cooling constant k and used it to predict the temperature of the coffee at any time t. We have also graphed the cooling of the coffee to visualize the underlying physics.
References
- Newton, I. (1701). Opticks: Or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light.
- Crank, J. (1975). The Mathematics of Diffusion. Oxford University Press.
Further Reading
- Crank, J. (1984). Free and Moving Boundary Problems. Oxford University Press.
- Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids. Oxford University Press.
The Temperature of a Cup of Coffee Over Time: A Mathematical Analysis - Q&A ====================================================================
Introduction
In our previous article, we explored the mathematical concepts that govern the cooling of a cup of coffee over time. We used Newton's Law of Cooling to model the cooling of the coffee and estimated the value of the cooling constant k. In this article, we will answer some frequently asked questions about the cooling of the coffee.
Q: What is Newton's Law of Cooling?
A: 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:
dT/dt = -k(T - Ts)
where:
- T is the temperature of the object at time t
- Ts is the temperature of the surroundings
- k is the cooling constant
Q: How do you estimate the value of the cooling constant k?
A: To estimate the value of k, we can use the data from the table to substitute into the equation above. For example, at time t = 10 minutes, the temperature of the coffee is 180°F. We can substitute this value into the equation above to get:
180 = 70 + (200 - 70)e^(-k(10))
Simplifying the equation, we get:
110 = 130e^(-10k)
Dividing both sides by 130, we get:
0.846 = e^(-10k)
Taking the natural logarithm of both sides, we get:
-0.173 = -10k
Dividing both sides by -10, we get:
k = 0.0173
Q: How do you model the cooling of the coffee over time?
A: To model the cooling of the coffee over time, we can use the equation above:
T(t) = 70 + (200 - 70)e^(-0.0173t)
We can substitute the value of k into the equation to get:
T(t) = 70 + (200 - 70)e^(-0.0173t)
Q: How do you graph the cooling of the coffee?
A: To visualize the cooling of the coffee, we can graph the equation above. We can use a graphing calculator or software to plot the temperature of the coffee over time.
Q: What are some real-world applications of Newton's Law of Cooling?
A: Newton's Law of Cooling has many real-world applications, including:
- Cooling of buildings and homes
- Cooling of electronic devices
- Cooling of food and drinks
- Cooling of pharmaceuticals
Q: What are some limitations of Newton's Law of Cooling?
A: Newton's Law of Cooling assumes that the temperature of the surroundings is constant and that the object is cooling uniformly. In reality, the temperature of the surroundings may vary and the object may not cool uniformly.
Q: How do you account for the limitations of Newton's Law of Cooling?
A: To account for the limitations of Newton's Law of Cooling, we can use more complex models that take into account the variation in temperature of the surroundings and the non-uniform cooling of the object.
Conclusion
In this article, we have answered some frequently asked questions about the cooling of a cup of coffee over time. We have used Newton's Law of Cooling to model the cooling of the coffee and estimated the value of the cooling constant k. We have also discussed some real-world applications and limitations of Newton's Law of Cooling.
References
- Newton, I. (1701). Opticks: Or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light.
- Crank, J. (1975). The Mathematics of Diffusion. Oxford University Press.
Further Reading
- Crank, J. (1984). Free and Moving Boundary Problems. Oxford University Press.
- Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids. Oxford University Press.