Problem 4 - Active Calculus

Question

Newton's Law of Cooling says that the rate at which an object, such as a cup of coffee, cools is proportional to the difference in the object's temperature and room temperature. If \(T(t)\) is the object's temperature and \(T_{r}\) is room temperature, this law is expressed at $$ \frac{d T}{d t}=-k\left(T-T_{r}\right) $$ where \(k\) is a constant of proportionality. In this problem, temperature is measured in degrees Fahrenheit and time in minutes. a. Two calculus students, Alice and Bob, enter a \(70^{\circ}\) classroom at the same time. Each has a cup of coffee that is \(100^{\circ} .\) The differential equation for Alice has a constant of proportionality \(k=0.5,\) while the constant of proportionality for Bob is \(k=0.1 .\) What is the initial rate of change for Alice's coffee? What is the initial rate of change for Bob's coffee? b. What feature of Alice's and Bob's cups of coffee could explain this difference? c. As the heating unit turns on and off in the room, the temperature in the room is $$ T_{r}=70+10 \sin t $$ Implement Euler's method with a step size of \(\Delta t=0.1\) to approximate the temperature of Alice's coffee over the time interval \(0 \leq t \leq 50 .\) This will most easily be performed using a spreadsheet such as Excel. Graph the temperature of her coffee and room temperature over this interval. d. In the same way, implement Euler's method to approximate the temperature of Bob's coffee over the same time interval. Graph the temperature of his coffee and room temperature over the interval. e. Explain the similarities and differences that you see in the behavior of Alice's and Bob's cups of coffee.

Step-by-Step Solution

Answer

Alice's coffee cools at -15°F/min initially, while Bob's cools at -3°F/min. Alice's faster cooling rate is due to her higher proportionality constant, leading to a quicker temperature change compared to Bob's coffee.

Step 1: Understanding Newton's Law of Cooling

Newton's Law of Cooling is given by the differential equation: \[ \frac{d T}{d t} = -k (T - T_{r}) \]In this equation, \(T(t)\) represents the temperature of the object at time \(t\), \(T_{r}\) is the room temperature, and \(k\) is the constant of proportionality.

Step 2: Finding Initial Rate of Change for Alice and Bob

To find the initial rate of change for Alice's and Bob's coffee, we substitute the given values into the differential equation.For Alice: \( T = 100^{\text{o}} \text{F}, T_{r} = 70^{\text{o}} \text{F}, k = 0.5 \).Thus,\[ \frac{dT}{dt} \bigg|_{t=0} = -0.5 (100 - 70) = -0.5 \times 30 = -15^{\text{o}} \text{F/min} \]For Bob: \( T = 100^{\text{o}} \text{F}, T_{r} = 70^{\text{o}} \text{F}, k = 0.1 \).Thus,\[ \frac{dT}{dt} \bigg|_{t=0} = -0.1 (100 - 70) = -0.1 \times 30 = -3^{\text{o}} \text{F/min} \]

Step 3: Explaining the Difference in Cooling Rates

The difference in the initial cooling rates for Alice's and Bob's coffee can be attributed to the different values of the constant \(k\). Alice's coffee cools faster because the constant of proportionality \(k = 0.5\) is higher compared to Bob's \(k = 0.1\).

Step 4: Euler's Method for Alice's Coffee Temperature

Using Euler's method with a step size of \( \, \, t = 0.1 \), the temperature of Alice's coffee can be approximated over the time interval \(0 \, \leq \, t \, \leq \, 50\). The formula for Euler's method is:\[ T_{n+1} = T_{n} + \frac{dT}{dt} \bigg|_{T=T_{n}} \, \Delta t \]Where \( \, \Delta t = 0.1 \, \).Set up a spreadsheet with columns for \(t\) values (incrementing by 0.1), \(T\) values for Alice's coffee, and \(T_{r} = 70 + 10 \, \sin t\). Use the initial condition \(T = 100^{\text{o}} \text{F}\). Perform the calculations iteratively and graph the results.

Step 5: Euler's Method for Bob's Coffee Temperature

Similarly, implement Euler's method to approximate the temperature of Bob's coffee over the time interval \(0 \, \leq \, t \, \leq \, 50\) using the same step size \( \, \Delta t = 0.1 \). Set up a spreadsheet with similar columns (\(t, T\), and \(T_{r}\)), and use initial condition \(T = 100^{\text{o}} \text{F}\). Perform the calculations iteratively and graph the results.

Step 6: Comparing the Results

Graph both the temperature of Alice's and Bob's coffee along with the room temperature over the interval \(0 \, \leq \, t \, \leq \, 50\). Observe that Alice's coffee cools down more rapidly than Bob's due to the higher value of \(k\), making her differential equation more responsive to the temperature difference from the room temperature.

Key Concepts

Differential Equations

Differential equations are mathematical equations that describe how a quantity changes in relation to another quantity. In the context of Newton's Law of Cooling, the differential equation used is: \[ \frac{dT}{dt} = -k (T - T_r) \] This equation shows the rate of temperature change (\frac{dT}{dt}) of an object over time. Here, T(t) is the current temperature of the object, T_r is the ambient or room temperature, and k is the constant of proportionality. Differential equations are crucial in modeling natural phenomena, including cooling, population growth, and other dynamic processes. Understanding these equations can help predict behavior and make informed decisions in various fields.

Euler's Method

Euler's Method is a numerical technique to solve differential equations. It allows us to approximate the solution by stepping forward a small interval at a time. The formula for Euler's Method is: \[ T_{n+1} = T_n + \frac{dT}{dt} \bigg|_{T=T_n} \, \Delta t \] Here, T_n represents the current temperature at step n, and \, \Delta t \, is the step size. For example, if we start with an initial temperature and use a step size of 0.1 minutes, we can calculate the next temperature by following the equation iteratively.

• Set up a table with columns for time (t), temperature (T), and room temperature (T_r).
• Use the initial temperature as the starting value (T_0).
• Calculate the new temperature using the formula and update the table.
• Repeat the process for each time step.

This method is especially useful when analytical solutions are challenging or impossible to obtain.

The Proportionality Constant in Cooling

In Newton's Law of Cooling, the constant of proportionality, k, plays a significant role in determining the rate at which an object cools. The differential equation is: \[ \frac{dT}{dt} = -k (T - T_r) \] The value of k indicates how quickly the temperature difference between the object and the surroundings diminishes.

• A higher value of k means the object cools faster.
• A lower value of k means the object cools more slowly.

For instance, in our example, Alice's coffee has a k value of 0.5, leading to a faster cooling rate compared to Bob's coffee with a k value of 0.1. This explains why Alice's coffee cools at -15°F per minute initially, while Bob's coffee cools at -3°F per minute. High k values might correspond to better heat conductivity or a greater surface area exposed to the ambient environment, leading to quicker cooling.