Problem 14 - Thomas Calculus

Question

Mixture problem \(A 200\) -gal tank is half full of distilled water. At time \(t=0,\) a solution containing 0.5 lb/ gal of concentrate enters the tank at the rate of 5 gal/min, and the well-stirred mixture is withdrawn at the rate of 3 gal \(/\) min. a. At what time will the tank be full? b. At the time the tank is full, how many pounds of concentrate will it contain?

Step-by-Step Solution

Answer

a. 50 minutes; b. 125 pounds.

Step 1: Determine the Initial Condition

At the start, the tank contains 100 gallons of water because it is half full (200 gallons / 2). This will be the starting volume of the liquid in the tank when a new solution begins entering.

Step 2: Calculate the Rate of Change in Volume

The solution enters at a rate of 5 gal/min and exits at 3 gal/min. Thus, the net rate of change in volume is \(5 - 3 = 2\) gallons per minute. This means the tank gains 2 gallons per minute.

Step 3: Determine When the Tank Becomes Full

Start with 100 gallons and add \(2\) gallons per minute. We need to find when the tank will reach 200 gallons.Set up the equation for time: \[100 + 2t = 200\]Solving for \(t\), we get \[2t = 100\]\[t = 50\] minutes. Thus, it will take 50 minutes to fill the tank.

Step 4: Establish the Concentrate's Rate of Entry

The solution entering the tank has a concentration of 0.5 lb/gal. Since it enters at 5 gal/min, the rate of entry for concentrate is \[0.5 \times 5 = 2.5\] lbs/min.

Step 5: Calculate the Amount of Concentrate When the Tank is Full

We need to find the amount of concentrate added in 50 minutes.Multiply the rate of concentrate by the time it takes to fill the tank:\[2.5 \times 50 = 125\] pounds.

Key Concepts

Differential Equations

Differential equations are a crucial tool in solving mixture problems because they describe how the quantity of a substance within a system changes over time. In the context of this problem, the differential equation would model how the concentration of the solution in the tank changes as more solution flows in and out. When we analyze such scenarios, a key goal is to understand how these rates contribute to changes in the mixture's concentration and volume. In mathematical terms, a differential equation can help express how the total volume of water and the amount of concentrate adjust when new solution enters the tank, ensuring the system is in constant flux.
The general form of a differential equation for this mixture problem could involve both the input rate of concentrate and the output rate of the mixture, making it possible to use calculus to find exact solutions for varying concentrations at different times.

Rate of Change

The rate of change is another essential concept in understanding how the conditions of the tank change over time. In this exercise, you can think of the rate of change in terms of both volume and concentration.

• The rate of change in volume: This is calculated by subtracting the rate at which liquid leaves the tank from the rate at which it enters, resulting in the net inflow rate of 2 gallons per minute.
• The rate of change in concentrate: This refers to how quickly the concentrate enters the tank, expressed in pounds per minute. Here, concentrate enters the tank at a rate of 2.5 lbs/minute based on the supplied concentration and flow rates.

Understanding these rates is vital as it allows us to form equations that help predict the conditions at any given time, like when the tank will be full or how much concentrate it will contain.

Initial Conditions

Initial conditions provide the starting point for analyzing how a system evolves over time. They are crucial in mixture problems, as they set the stage for any calculations involving differential equations and rates of change.
In this problem, the initial condition tells us that the tank begins with 100 gallons of water, given that it is half full. This initial state is necessary for determining how the volume in the tank increases—and ultimately when it will become full. By knowing these conditions, it becomes easier to set up the problem, establish the current state of the tank, and predict future outcomes.
Initial conditions help form the baseline of the differential equations used, offering an effective "starting line" from which all subsequent changes are measured. This enables accurate and efficient solutions that address both volumes and concentrations as the scenario progresses.