Problem 19 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A canister contains 10 liters of blue paint. Paint is being used at a rate of 2 liters per hour and the canister is being replenished at a rate of 2 liters per hour by a pale blue paint that is \(80 \%\) blue and \(20 \%\) white. Assuming the canister is well-mixed, write a differential equation whose solution is \(w(t)\), the amount of white paint in the canister at time \(t\). Specify the initial condition.

Step-by-Step Solution

Answer

The initial condition is: \(w(0) = 0\). The differential equation is: \(\frac{dw}{dt} = - \frac{2w}{10} + 0.4\).

Step 1: Determine the Initial Conditions

The initial conditions are taken directly from the problem statement. We know initially, the canister only contains blue paint. Since the problem statement asks for \(w(t)\), the amount of white paint, the initial condition is \(w(0) = 0\)

Step 2: Modeling the Gains and Losses

Let's analyze the changes happening in the canister. The white paint is leaving the canister mixed with the blue paint at a rate of 2 liters per hour, but it is also entering the canister mixed in the pale blue paint at the same rate. Since the pale blue paint is 80% blue and 20% white, for every liter of pale blue paint, 0.2 liters is white paint. And thus, for every hour, 2 liters * 0.2 = 0.4 liters of white paint is being added to the canister. We can write these rates as follows:\n\nRate of white paint leaving the canister = - \(2w/10\) (2 liters per hour that's \(w/10\) part white) \nRate of white paint entering the canister = + 0.4 (0.2 liters of white paint per liter of pale blue paint * 2 liters entering per hour)

Step 3: Formulate the Differential Equation

Combining the rates of gain and loss, we can write a differential equation for the amount of white paint in the canister as a function of time \(t\). The rate of change of white paint with respect to time is given by the equation \n\n\(\frac{dw}{dt} = - \frac{2w}{10} + 0.4\)

Key Concepts

Initial Conditions

Initial conditions are crucial in solving differential equations because they establish the starting point for the problem. They allow us to uniquely determine the solution to the differential equation. In this context, they tell us the amount of white paint at the beginning, when time, denoted as \( t \), is zero.

From the exercise, we understand that initially, the canister is full of blue paint. Therefore, at time \( t = 0 \), the quantity of white paint in the canister, \( w(0) \), is zero. This initial condition is a fundamental piece of information as it anchors our solution to a known starting point.

Whenever you're dealing with initial conditions, remember it's like setting up a story. You need to know where the story begins to understand how it unfolds, especially in dynamic processes captured by differential equations.

Rate of Change

In differential equations, the rate of change of a quantity with respect to time is a core concept. The rate of change is often represented as a derivative, such as \( \frac{dw}{dt} \) in our problem.

This derivative tells us how the amount of white paint, \( w \), changes over time \( t \). In the problem, the change happens due to two simultaneous processes:

• White paint is leaving the canister as part of the well-mixed solution at a certain rate.
• New white paint is being added to the canister at a different rate.

The overall rate of change in the amount of white paint can be found by subtracting the rate it leaves from the rate it enters.

Mathematically, this is given by the differential equation \( \frac{dw}{dt} = -\frac{2w}{10} + 0.4 \). Here, \(-\frac{2w}{10} \) captures the loss, while \(0.4\) captures the gain per hour. This formula balances the inflow and outflow to reveal how \( w \) evolves over time.

Modeling

Modeling with differential equations allows us to translate real-world scenarios into mathematical frameworks. It involves capturing the essence of a process using equations that depict changes and dynamics over time.

In the given exercise, the task was to model the amount of white paint in a canister over time. The model must include how paint is added and removed from the canister, considering the properties of what is added. Here, the pale blue paint added contains 20% white paint. Therefore, for every liter added, 0.2 liters is white paint.

• The model considers the inflow and outflow of white paint.
• The inflow rate is 0.4 liters per hour based on the 0.2 liters per liter ratio and a 2-liter per hour replenishment rate.
• The outflow rate considers the concentration of white paint in the mixed canister, losing \( \frac{2w}{10} \) liters per hour.

By setting up a differential equation, \( \frac{dw}{dt} = -\frac{2w}{10} + 0.4 \), we encapsulate the entire process of paint mixing and changing in the canister. This equation serves as a mathematical model to predict the future dynamics of the paint composition.