Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A wet dish towel is put on the back of a kitchen chair to dry. It dries at a rate proportional to the difference in moisture content between the dishtowel and the kitchen air. Assume that the moisture content in the air is fixed and is given by \(M\). (a) Set up the differential equation involving \(W=W(t)\), the amount of water in the dish towel at time \(t\). (b) Find and sketch the solution.

Step-by-Step Solution

Answer

The resulting differential equation is \(W'(t) = -k*(W(t) - M)\) and its solution is \(W(t) = Ce^{-kt} + M\). The graph of \(W(t)\) shows exponential decay or growth towards \(M\), the fixed moisture content in the air.

Step 1: Formulate the differential equation

First, start with the sentence 'It dries at a rate proportional to the difference in moisture content between the dish towel and the air.' This can be translated into a differential equation. The rate of change of the water content \(W\) of the dish towel can be represented by \( -k*(W - M)\). The negative sign represents that the water content of the dish towel is decreasing over time. So, the differential equation becomes \(W'(t) = -k*(W(t) - M)\).

Step 2: Solve the differential equation

To solve this first order linear differential equation, we integrate on both sides. This results in the solution \(W(t) = Ce^{-kt} + M\), where \(C\) is the integration constant, which can be determined by initial conditions.

Step 3: Sketch the solution

Now, for sketching the solution consider the two different scenarios. If the moisture content in the dishtowel is initially more than \(M\) (i.e. in the beginning), the graph will show a decreasing function converging to \(M\). If at \(t=0\), \(W(t) < M\), the graph will show an increasing function converging to \(M\). This is because with time, the moisture content in the towel and the air will tend to equalize.

Key Concepts

First-order Linear Differential Equation

A first-order linear differential equation is a type of equation that involves the derivative of a function and the function itself. It has the standard form:

• \( y'(t) + p(t)y(t) = g(t) \)

In our exercise, the function describes how the water content in a dish towel changes over time. The specific equation is:

• \(W'(t) = -k(W(t) - M)\)

This form shows that the rate of change of water content is proportional to its current value minus the moisture content of the air, which is a fixed constant.
The term "linear" signifies that both the dependent variable, here \( W \), and its derivative appear only raised to the first power. Notice the negative sign: it indicates that as time progresses, the amount of moisture in the towel decreases because the towel becomes drier over time.

Rate of Change

The rate of change is a fundamental concept in calculus that describes how a quantity evolves over time or another variable. In this context, the rate of change of the water content in the dish towel reflects how quickly it loses moisture. It is expressed as the derivative of the water content, \( W'(t) \).
The exercise describes that this rate is "proportional to the difference in moisture content." Mathematically, this is depicted by multiplying a constant \( -k \) by the difference \((W - M)\). Here, \( k \) is a positive constant that dictates how fast the drying process occurs:

• A larger \( k \) implies faster drying.
• A smaller \( k \) suggests slower drying rates.

Proportional relationships are pivotal in understanding real-world processes, like how fast a wet item dries when exposed to air of a different moisture content.

Moisture Content

Moisture content, in this exercise, refers to how much water is present in the dish towel and the air around it. The symbol \( W(t) \) is used to denote the moisture content in the towel at a certain time \( t \). Meanwhile, \( M \) represents the constant moisture content of the surrounding air.
The drying process occurs because of the difference between \( W \) and \( M \). If \( W \) is greater than \( M \), it means the towel is wetter than the air. As time passes, \( W(t) \) decreases towards \( M \) as the towel loses moisture.

• If initially, the towel is very wet, there is a large difference, leading to a faster rate of drying.
• If the towel is less wet initially, the drying rate is slower.

Understanding moisture content helps grasp the concept of equilibrium, where after a long time, the towel's moisture becomes akin to that of the surrounding air.

Initial Conditions

Initial conditions are crucial for solving differential equations as they provide specific information to determine the unique solution. For the given problem, the initial condition is the moisture content at the start of the observation, typically denoted as \( W(0) \).
By integrating the differential equation, we get the general solution \( W(t) = Ce^{-kt} + M \). Here, \( C \) is an integration constant that must be defined using the initial condition. By substituting \( t = 0 \) and the initial moisture content \( W(0) \) into the equation, we can solve for \( C \).

• If \( W(0) > M \), the solution describes a decreasing curve towards equilibrium at moisture \( M \).
• If \( W(0) < M \), it outlines an increasing curve towards \( M \) as moisture is absorbed from the air.

Initial conditions personalize the model by relating it to specific starting scenarios, allowing us to predict more precisely the towel's drying behavior over time.