Problem 68

Question

A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. Show that the concentration of salt after $ t $ minutes (in grams per liter) is $$ C(t) = \frac{30t}{200 + t} $$ (b) What happens to the concentration as $ t \to \infty $?

Step-by-Step Solution

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Answer

As $ t \to \infty $, the concentration $ C(t) \to 30 $ g/L.

1Step 1: Understand the Problem

We need to find the concentration of salt in the tank after $ t $ minutes given a brine solution (30 g/L) is added at 25 L/min. Initially, the tank contains 5000 L of pure water and no salt.

2Step 2: Define Variables

Let $ S(t) $ be the amount of salt in grams in the tank at time $ t $. The rate at which salt is added to the tank is $ R_{in} = 25\, \text{L/min} \times 30\, \text{g/L} = 750\, \text{g/min} $.

3Step 3: Construct Differential Equation

The rate of change of salt $ \frac{dS}{dt} $ is equal to the rate at which salt is added since initially there's only pure water. Thus: $ \frac{dS}{dt} = 750 $.

4Step 4: Solve the Differential Equation

Integrate the differential equation $ \frac{dS}{dt} = 750 $ to find $ S(t) $. Integrating both sides gives $ S(t) = 750t + C_0 $, where $ C_0 $ is the integration constant. Initially, $ S(0) = 0 $, so $ C_0 = 0 $. Thus, $ S(t) = 750t $.

5Step 5: Determine Total Volume of Liquid

The total volume of liquid in the tank at time $ t $ is the initial 5000 L plus the 25 L/min that is added. Thus, $ V(t) = 5000 + 25t $.

6Step 6: Calculate Concentration of Salt

The concentration $ C(t) $ in grams per liter is the amount of salt divided by the volume of liquid: $ C(t) = \frac{S(t)}{V(t)} = \frac{750t}{5000 + 25t} $. Simplifying this gives $ C(t) = \frac{30t}{200 + t} $.

7Step 7: Analyze Behavior as $ t \to \infty $

As $ t \to \infty $, the concentration $ C(t) = \frac{30t}{200 + t} $ approaches 30 g/L because the $ t $ terms dominate and the constant terms become insignificant.

Key Concepts

Concentration CalculationsBrine SolutionRate of ChangeSalt Concentration

Concentration Calculations

Concentration calculations are handy when you need to determine how much of a particular substance is present in a solution. In this scenario, concentration is derived from the amount of salt dissolved in a given volume of liquid. We start with pure water. Over time, brine, which is a salty liquid, is added. The brine has a salt concentration of 30 g/L.
To find the salt concentration at a particular time, you need to:

Calculate the total amount of salt added
Determine the total volume of the solution at the given time
Divide the total salt amount by the total volume

This method helps us find how concentrated the solution is during any given moment.

Brine Solution

A brine solution is simply water saturated with salt. In this exercise, the brine contains 30 grams of salt per liter. This specific concentration is important, as it affects how quickly the concentration of salt changes in the tank.
Understanding the properties of brine is crucial here:

Brine is denser than pure water due to the dissolved salt.
The concentration in this problem is expressed in grams per liter (g/L).
As brine flows into the tank, it increases the tank's overall salt content.

Knowing these factors allows us to predict how the concentration of salt develops over time as brine is continually added.

Rate of Change

The rate of change is a measure of how quickly something happens. In this problem, it refers to how fast salt is being added to the tank. We express this change using a differential equation.
For this problem:

The rate of salt added to the tank is 750 grams per minute, calculated by multiplying 30 g/L by 25 L/min.
The differential equation, $ \frac{dS}{dt} = 750 $, shows the relationship between time and the amount of salt.
When solving the problem, we integrate the equation to find the total amount of salt at any time $ t $.

This approach helps us track how salt concentration builds over time as more salt is added.

Salt Concentration

Salt concentration provides a measure of the amount of salt within a solution. In the context of this problem, it's essential to understand how the concentration evolves as time progresses. To determine salt concentration:

Use the formula $ C(t) = \frac{30t}{200 + t} $ derived from dividing the salt amount by the total liquid volume.
Initially, since the tank starts with pure water, the concentration is zero.
As time goes to infinity, $ t \to \infty $, the concentration approaches 30 g/L, meaning the solution's salt concentration aligns with that of the brine being added.

This formula and understanding the limit behavior help to predict the outcome as time continues to pass.

Problem 67

Problem 68

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