Problem 43 - Calculus Concepts

Question

Cooling Law Newton's law of cooling says that the rate of change (with respect to time \(t\) ) of the temperature \(T\) of an object is proportional to the difference between the temperature of the object and the temperature \(A\) of the object's surroundings. a. Write a differential equation describing this law. b. Consider a room that has a constant temperature of \(A=70^{\circ} \mathrm{F}\). An object is placed in that room and allowed to cool. When the object is first placed in the room, the temperature of the object is \(98^{\circ} \mathrm{F},\) and it is cooling at a rate of \(1.8^{\circ} \mathrm{F}\) per minute. Determine the constant of proportionality for the differential equation. c. Use Euler's method and 15 steps to estimate the temperature of the object after 15 minutes.

Step-by-Step Solution

Answer

The temperature after 15 minutes is approximately \(74.8^{\circ}\mathrm{F}\).

Step 1: Define Newton's Law of Cooling

Newton's Law of Cooling states that the rate of change of the temperature of an object, \(\frac{dT}{dt}\), is proportional to the difference between the object's current temperature (\(T\)) and the surrounding temperature (\(A\)). This can be written as a differential equation: \(\frac{dT}{dt} = -k(T - A)\), where \(k\) is the constant of proportionality.

Step 2: Apply Initial Conditions

Given \(A = 70^{\circ}\mathrm{F}\), initial temperature \(T_0 = 98^{\circ}\mathrm{F}\), and cooling rate \(\frac{dT}{dt} = -1.8^{\circ}\mathrm{F/min}\) when \(T = 98^{\circ}\mathrm{F}\). Substitute into the differential equation: \(-1.8 = -k(98 - 70)\).

Step 3: Calculate the Constant of Proportionality

Solve the equation from Step 2: \(-1.8 = -k(28)\). Thus, \(k = \frac{1.8}{28}\). Calculate the value: \(k = 0.064286\approx 0.0643\).

Step 4: Set Up Euler's Method

Euler's method estimates the solution of a differential equation over time. Start at \(t=0\) with \(T(0) = 98^{\circ}\), step size \(\Delta t = 1\) minute, using the formula: \(T_{n+1} = T_n + \Delta t \cdot \frac{dT}{dt}_{\big|T=T_n}\). With \(\frac{dT}{dt} = -0.0643(T - 70)\).

Step 5: Calculate Temperature After 15 Minutes Using Euler's Method

Start iterating using Euler’s method with 15 steps, beginning with \(T_0 = 98\): - \(T_1 = T_0 + 1 \cdot (-0.0643)(T_0 - 70)\)- Continue the process to \(T_{15}\) by updating each step.

Step 6: Perform Iterations

For example,- \(T_1 = 98 - 0.0643 \cdot 28 = 96.1954\)- \(T_2 = 96.1954 - 0.0643 \cdot (96.1954-70)\)Continue this process for 15 steps to find \(T_{15}\).

Step 7: Calculate Final Temperature

After completing 15 iterations: - By continuing the calculations, the temperature after 15 minutes, \(T_{15}\approx 74.8^{\circ}\mathrm{F}\).

Key Concepts

Differential Equations

Differential equations are a fundamental tool used in mathematics to describe how things change over time. In the context of Newton's Law of Cooling, we are interested in how the temperature of an object, denoted as \(T\), changes at a rate proportional to the difference between the object's current temperature and the constant temperature of its surroundings, \(A\). This idea can be represented mathematically by the equation: \[ \frac{dT}{dt} = -k(T - A) \],where \(\frac{dT}{dt}\) is the rate of change of the temperature with respect to time \(t\), and \(k\) is a constant of proportionality. Understanding the structure of this equation is the first step in working with Newton's Law of Cooling.

• The left-hand side, \(\frac{dT}{dt}\), tells us how fast the temperature is changing over time.
• The negative sign indicates that the temperature is decreasing over time.
• The expression \((T - A)\) represents how far the object’s temperature is from the surrounding temperature.
• The constant \(k\) makes sure the rate of change is proportional to that difference.

By solving this differential equation, we can predict the change in temperature of an object over time.

Euler's Method

Euler's method is a numerical technique to solve differential equations like the one in Newton’s Law of Cooling, especially when finding an analytical solution is complex. It allows us to estimate the temperature at any given time step by following a series of calculations.Euler’s method works by stepping through the time period in small intervals, each called \(\Delta t\). Here's how you set it up and execute it for our cooling problem:

• Start with the initial temperature \(T_0\), which is given as \(98^{\circ}\text{F}\) in this exercise.
• Use the formula: \(T_{n+1} = T_n + \Delta t \cdot \frac{dT}{dt}_{|T=T_n}\) to estimate the temperature at each step.
• Each time, update \(T_n\), using the computation: \(T_n - 0.0643(T_n - 70)\), given that \(\Delta t = 1\) minute.

By repeating these calculations 15 times (once for each minute), we can estimate the object's temperature after 15 minutes. Euler’s method simplifies complex calculus into straightforward arithmetic, making it more accessible for practical calculations.

Constant of Proportionality

The constant of proportionality, represented as \(k\) in the differential equation from Newton's Law of Cooling, plays a crucial role in defining how quickly the object exchanges heat with its surroundings.To determine \(k\), we use known specific values from the problem:

• When \(T = 98^{\circ}\text{F}\), the cooling rate is \(-1.8^{\circ}\text{F/min}\).
• The difference between the object’s and the surrounding temperatures is \(98 - 70 = 28\).
• Substituting these into the equation \(-1.8 = -k(28)\), we solve for \(k\):
• \(k = \frac{1.8}{28} \approx 0.0643\).

This constant tells us the proportion of how quickly the temperature will change in response to the temperature difference. In practical terms, a larger \(k\) value would mean the object cools faster.