Question

A programmable calculator or computer is needed for these exercises. Assume a tank having a capacity of 200 gal initially contains 90 gal of fresh water. At time \(t=0\), a salt solution begins flowing into the tank at a rate of \(6 \mathrm{gal} / \mathrm{min}\) and the well-stirred mixture flows out at a rate of 1 gal/min. Assume that the inflons concentration is given by \(c(t)=2-\cos \pi t\) oz/gal, where time \(t\) is in minutes. Formulate the appropriate initial value problem for \(Q(t)\), the amount of salt (in ounces) in the tank at time \(t\). Use Euler's method to approximately determine the amount of salt in the tank when the tank contains 100 gal of liquid. Use a step size of \(h=0.01\).

Short answer

Question: Use Euler's method to approximate the amount of salt in a tank when it contains 100 gal of liquid, given the initial value problem for the rate of change of salt in the tank and a step size of h=0.01. Solution: By implementing Euler's method with step size h=0.01 and iterating from t=0 to t=2, you should obtain an approximate value of the amount of salt in the tank, Q(2), when it contains 100 gal of liquid.

Step-by-step solution

Find the balance equation for the rate of change of salt in the tank

We can start by finding the balance equation for the rate of change of salt in the tank. The inflow rate is given by the product of the flow rate (6 gal/min) and the concentration \(c(t)\). The outflow rate is given by the product of the flow rate (1 gal/min) and the concentration of salt in the tank, which is \(\frac{Q(t)}{90+t}\). Therefore, the rate of change of salt in the tank can be written as: \(\frac{dQ}{dt} = 6c(t) - \frac{Q(t)}{90+t}\)

Formulate the initial value problem

The initial value problem for the amount of salt in the tank is to find \(Q(t)\) given the differential equation and the initial condition. At time \(t=0\), the tank initially contains 90 gal of fresh water, which means there is no salt present. So, the initial condition is \(Q(0) = 0\).

Implement Euler's method

To use Euler's method with a step size of \(h=0.01\), we will iterate the following formula: \(Q(t+h) = Q(t) + h \left( 6c(t) - \frac{Q(t)}{90+t} \right)\) We will stop iterating when the amount of liquid in the tank reaches 100 gal. Since liquid is flowing in at a rate of 6 gal/min and flowing out at a rate of 1 gal/min, the net inflow rate is 5 gal/min. Therefore, it will take \(\frac{100-90}{5}\) = 2 minutes for the tank to contain 100 gal of liquid. So, we will iterate from \(t=0\) to \(t=2\) with a step size of \(h=0.01\).

Calculate the amount of salt in the tank when it contains 100 gal of liquid

Iterate through the Euler's method formula and find the amount of salt in the tank at \(t=2\). 1. Initialize the variables: \(Q(0)=0\), \(t=0\), \(h=0.01\), and \(T=2\). 2. While \(t < T\): a. Calculate \(c(t) = 2 - \cos\pi t\) b. Update \(Q(t)\): \(Q(t+h) = Q(t) + h \left( 6c(t) - \frac{Q(t)}{90+t} \right)\) c. Update the time: \(t = t + h\) 3. The final value of \(Q(t)\) is the amount of salt in the tank when it contains 100 gal of liquid. By implementing these steps, you should obtain a value for \(Q(2)\) which represents the amount of salt in the tank when it contains 100 gal of liquid.

Key concepts

Understanding Initial Value Problems

In mathematics, an initial value problem refers to a type of differential equation along with a specific value called the initial condition at a given point in time. The goal here is to find a function that not only satisfies the differential equation but also meets the designated initial condition.

In the context of the given exercise, we are dealing with the initial value problem of finding the amount of salt Q(t) in a tank over time, given the rate of change of salt in the tank and that at time t=0, the tank contains a known volume of fresh water, hence an initial salt amount of Q(0) = 0. Solving this problem provides insight into how processes evolve over time in response to various rates and initial conditions.

Rate of Change of Salt Concentration

The rate of change of salt concentration in a tank is an essential aspect of many processes. It gives us the change in concentration of a substance (in this case, salt) over time within a closed system. This rate is usually affected by factors such as the inflow and outflow rates and the concentration of solute in these flows.

For the problem we have, the rate of change of the amount of salt Q(t) is determined by the combined effect of the salt solution flowing in and the well-stirred mixture flowing out of the tank. Mathematically, this can be represented by the balance equation which ties together the rate of change with these influencing variables.

Numerical Integration via Euler's Method

Numerical integration methods are techniques used to approximate the solutions to differential equations when analytical solutions are difficult or impossible to obtain. Euler's Method is a straightforward numerical procedure for solving initial value problems with a given step size.

To apply Euler's method, we proceed from the initial condition by small increments or 'steps' along the time axis, using the slope of the tangent to the curve (given by the differential equation) to estimate the increase in our function. It's a sort of numerical trial-and-error approach that gradually builds a solution by taking these regular steps from the known starting point to the desired point in time.

The Balance Equation Concept

A balance equation models the conservation of some quantities, such as mass or energy, across a system's boundaries. When it comes to changing solute concentration in a tank, the balance equation helps us understand how the combination of all inputs and outputs affects the total amount within the system.

In our scenario, the balance equation is derived from considering the volume of the inflow with its salt concentration and the volume of the outflow as a fraction of the current amount in the tank. This equation encapsulates the dynamic nature of the system and is the foundation upon which we base our application of Euler's method to approximate the amount of salt Q(t) at any given time.