{Use of Tech} A mixing tank A 500-liter (L) tank is filled wit...

Question

{Use of Tech} 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 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.
b. Graph the volume function and verify that the tank is empty when t=1000 min.

b. Graph the volume function and verify that the tank is empty when t=1000 min.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a reactor is initially filled with 800 L of a chemical solution. At T equal to 0, a reactant starts being added to the reactor at a rate of 4 L per minute, while the reaction mixer is removed at a rate of 4.4 L per minute. The volume of the solution in the reactor decreases over time and is represented by the function VFT is equal to 800 minus 0.4 T, where VFT is in liters and T is in minutes. We're asked to graph the volume function. So the volume function that we need to graph is our VFT function. And we noticed that this function is linear. So, all I really need to do is determine two points for this um equation, and use those to create a line on our graph. So, the first point we're going to look at is when T is equal to 0. So we will have V of 0. is equal to 800. Minus 0.4, multiplied by 0, which is equal to 800, which is also what we were told was the initial volume of our solution at T equal to 0 in our problem. Now, our volume here cannot be negative. So the smallest volume that I could have in this equation is 0. And so we're going to find out what value of T corresponds to V equal being equal to 0. So we're going to set V equal to 0, so we'll have 0, it's equal to 800. Minus 0.4, multiplied by T. So now I need to solve this equation for T. So, to isolate T, we're gonna add 0.4 T to both sides. And when we do that, We will get 0.4 T. is equal to 800. So now just need to divide both sides of the equation by 0.4. And doing so, I can solve for T and T is going to be equal to 2000. That would be 2000 minutes. So, that will give us our 2nd point that we need to create our line on our graph. So now we need to create our graph. And so we're going to be looking at a graph that has our volume plotted on the Y axis and our time plotted on the X axis. And so on a graph we'll need to mark two points. The first point is going to be uh when T is equal to 0, B is going to be equal to 800, and the second point is going to be when T is 2000. And V is equal to 0. So now all I'll need to do is actually connect those two points. So we'll connect those with a straight line. And once we've done that, this is the graph that the question was looking for. So this is the answer to the problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

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