A mixing tank A 500 -liter (L) tank is filled with pure...

A mixing tank A 500 -liter (L) tank is filled with pure water. At time $t=0,$ a salt solution begins flowing into the tank at a rate of $5 \mathrm{L} / \mathrm{min.}$ At the same time, the (fully mixed) solution flows out of the tank at a rate of $5.5 \mathrm{L} / \mathrm{min}$. The mass of salt in grams in the tank at any time $t \geq 0$ is given by $$M(t)=250(1000-t)\left(1-10^{-30}(1000-t)^{10}\right)$$ and the volume of solution in the tank is given by $$V(t)=500-0.5 t$$ a. Graph the mass function and verify that $M(0)=0$ b. Graph the volume function and verify that the tank is empty when $t=1000 \mathrm{min}$ c. The concentration of the salt solution in the tank (in $\mathrm{g} / \mathrm{L}$ ) is given by $C(t)=M(t) / V(t) .$ Graph the concentration function and comment on its properties. Specifically, what are $C(0)$ $\underset{t \rightarrow 1000^{-}}{\operatorname{and}} C(t) ?$ d. Find the rate of change of the mass $M^{\prime}(t),$ for $0 \leq t \leq 1000$ e. Find the rate of change of the concentration $C^{\prime}(t),$ for $0 \leq t \leq 1000$ f. For what times is the concentration of the solution increasing? Decreasing?

A mixing tank A 500 -liter (L) tank is filled with pure water. At time $t=0,$ a salt solution begins flowing into the tank at a rate of $5 \mathrm{L} / \mathrm{min.}$ At the same time, the (fully mixed) solution flows out of the tank at a rate of $5.5 \mathrm{L} / \mathrm{min}$. The mass of salt in grams in the tank at any time $t \geq 0$ is given by
$$M(t)=250(1000-t)\left(1-10^{-30}(1000-t)^{10}\right)$$
and the volume of solution in the tank is given by
$$V(t)=500-0.5 t$$
a. Graph the mass function and verify that $M(0)=0$
b. Graph the volume function and verify that the tank is empty when $t=1000 \mathrm{min}$
c. The concentration of the salt solution in the tank (in $\mathrm{g} / \mathrm{L}$ ) is given by $C(t)=M(t) / V(t) .$ Graph the concentration function and comment on its properties. Specifically, what are $C(0)$ $\underset{t \rightarrow 1000^{-}}{\operatorname{and}} C(t) ?$
d. Find the rate of change of the mass $M^{\prime}(t),$ for $0 \leq t \leq 1000$
e. Find the rate of change of the concentration $C^{\prime}(t),$ for $0 \leq t \leq 1000$
f. For what times is the concentration of the solution increasing? Decreasing?

Key Concepts

Application of the Product and Quotient Rules

When dealing with functions that involve the product or quotient of other functions, such as in expressions for mass or concentration, the product and quotient rules are essential tools in differentiation. These rules are used to accurately compute the derivatives needed to analyze the rates of change of these composite functions, thereby providing insights into the dynamic behavior of the system.

Concentration as a Derived Quantity

Concentration is determined by the ratio of the mass of a solute to the volume of the solvent. In mixing problems, the concentration function is derived from the mass and volume functions. Understanding this concept is important because it links the physical properties of the solution to its mathematical description, allowing for further analysis of how changes in the system, such as variations in inflow and outflow rates, affect overall concentration.

Analysis of Increasing and Decreasing Intervals

Determining whether a function is increasing or decreasing over a time interval requires analyzing the sign of its derivative. In mixing problems, this analysis helps identify when quantities like solute concentration are rising or falling. It provides a deeper understanding of the system's behavior over time and aids in decision-making processes, such as optimizing flow rates or predicting critical moments in the mixing process.

Rate of Change and Differentiation

Rate of change, typically found using derivatives, is a central idea in analyzing how a system evolves. In the context of mixing problems, it is used to determine how the mass of a solute or the concentration in a tank changes with time. Differentiation techniques allow us to find instantaneous rates of change, which are crucial in studying system behavior, optimizing processes, and predicting future states.

Mathematical Modeling of Mixing Processes

This concept involves creating mathematical representations of real-world mixing problems, where substances are added to and removed from a system. It uses conservation principles, such as mass balance, to set up differential equations that describe how quantities like mass and volume change over time. This modeling is fundamental for understanding the dynamics of concentration in systems where inflows and outflows occur simultaneously.

Graphical Analysis and Verification of Endpoints

Graphing functions provides a visual understanding of how a system behaves over its domain. By plotting functions like mass and volume with respect to time, one can verify essential conditions such as initial values and endpoints. This method helps in confirming that the mathematical model aligns with physical constraints, such as a tank eventually running empty or particular concentration levels being achieved.

Transcript

00:01 So let's start by looking at our mass function, which is m of t.

00:09 However, this graphing tool, it's better to use f of x.

00:13 So we'll call it f of x equals 250 times 1000 minus t, or 1 ,000 minus x actually, because our t value is x times 1 minus 10 to the negative 30 times 1 ,000.

00:42 Minus t to the tenth or one minus one thousand minus x to the tenth and then we close that off right there that's our first graph f of x and then what we have is our volume which is going to be 500 minus 0 .5 t so v of t equals 500 minus 0 .5 t and we're actually going to call it x and we're going to make this now g of x our main function i'm going to make these graphs show up a little better we're going to make the y value the y range much higher so we'll go from negative 500 to 500 and that should show us a little bit better values right here so that is our graph um and we can even make the x values smaller so we'll go from a negative one to a positive of 1.

02:00 That helps slightly.

02:04 So this is more what our graph is going to look like here.

02:08 Then what we want to do is graph the mass function and verify that m of 0 equals 0.

02:15 So we graph our mass function, which is this one, and we do see that m of 0 is 0.

02:21 Then we want to graph the volume function and verify that the tank is empty when t is equal to 1 ,000 minutes.

02:28 So now we'll graph this function and we see that at a thousand right here, it's equal to zero, which is what we wanted to verify.

02:41 And then we want to determine the concentration of the salt solution, which is going to be f of x divided by g of x.

02:52 So what we'll do is we'll have h of x equal to f of x divided by g of x.

03:01 This is our new graph which looks like this and we want to comment on its properties which is that it looks like a parabolic function but one that kind of levels out and stabilizes at this point right here which is starts to stabilize right here and then reaches zero gets slightly positive and then goes back down negative we want to know what c of zero is more specifically, and its value as it approaches negative, a positive 1 ,000 from the left.

03:46 So at 0 it equals 0, and then as it approaches 1 ,000 from the left, we see that it's undefined...

Please give Ace some feedback