Consider the two interconnected tanks shown in Figure 7.1 .6...

Consider the two interconnected tanks shown in Figure $7.1 .6 .$ Tank 1 initially contains 30 gal of water and 25 oz of salt, while Tank 2 initially contains 20 gal of water and 15 oz of salt. Water containing 1 oz/gal of salt flows into Tank 1 at a rate of 1.5 gal/min. The mixture flows from Tank 1 to Tank 2 at a rate of 3 gal/min. Water containing 3 oz/gal of salt also flows into Tank 2 at a rate of 1 gal/min (from the outside). The mixture drains from Tank 2 at a rate of 4 gal/min, of which some flows back into Tank 1 at a rate of 1.5 gal/min, while the remainder leaves the system. (a) Let $Q_{1}(t)$ and $Q_{2}(t),$ respectively, be the amount of salt in each tank at time $t .$ Write down differential equations and initial conditions that model the flow process. Observe that the system of differential equations is nonhomogeneous. (b) Find the values of $Q_{1}$ and $Q_{2}$ for which the system is in equilibrium, that is, does not change with time. Let $Q_{1}^{E}$ and $Q_{2}^{E}$ be the equilibrium values. Can you predict which tank will approach its equilibrium state more rapidly? (c) Let $x_{1}=Q_{1}(t)-Q_{1}^{E}$ and $x_{2}=Q_{2}(t)-Q_{2}^{E} .$ Determine an initial value problem for $x_{1}$ and $x_{2} .$ Observe that the system of equations for $x_{1}$ and $x_{2}$ is homogeneous.

Transcript

00:01 Okay, we have these tanks of salt water.

00:04 This first one has 30 gallons of water, and at any time t it has q1 ounces of salt in it.

00:11 At times zero, it has 25 ounces over here.

00:16 Then we have this other tank that initially has 20 gallons of water in it, and it has q2t of t ounces of salt.

00:24 So that changes, the amount of salt changes as the water comes in.

00:28 Initially it has 15 ounces so those two things are given there okay this one has salt water coming in from somewhere else salt water coming in from this tank over here and then it's also sending salt water out okay then this tank has water coming in oops are coming in up there and then it's sending water over to the other tank and it's sending water out onto the ground or in another somewhere else and the other tank is sending water to it.

01:05 Okay, so we want to make some differential equations that show the rate of change of salt per time.

01:14 Okay, so let's use black.

01:19 Dq1dt.

01:23 Okay, so let's start here at the top.

01:24 We got 1 .5 gallons per minute, gallons in one minute, times.

01:32 Times one ounce per gallon.

01:35 So that's going to give us ounces per minute.

01:39 Then we have this coming in.

01:42 It's coming in at 1 .5 gallons in one minute.

01:49 And it has q2 of t ounces of salt for 20 gallons.

02:00 And then it's sending out this mix right here.

02:07 So minus three gallons.

02:09 Per minute and how much salt is in it q1 t ounces per 30 gallons okay so that that's sort of the strength of it okay so let's get all the junk out of there so we got 1 .5 ounces per minute plus 1 .5 q2 over 20 minus 3 q1 over 30 so 1 .5 plus 15 over 200 so 3 over 40 q2 minus 1 tenth q1 so that's our first differential equation okay now we have to write one for the other one i need the picture though i'll scoot it just a little bit there we'll just have to move it up and now okay d q2 dt okay we have the fresh stuff coming in one gallon per minute three ounces per gallon, one gallon for one minute times three ounces per gallon.

03:29 Plus we have the water coming from the other tank, three gallons per minute, q1 ounces per 30 gallons, three gallons per minute, times q1 ounces for 30 gallons minus it's sending water out.

03:56 I'm going to do them separately.

04:02 Oops.

04:05 So we have, there we go.

04:12 So we have the one going this way.

04:15 Oops, up.

04:16 And it's going at 1 .5 gallons per minute.

04:25 And that one's strength is qt ounces per 20 gallon.

04:35 And is sending some this way 2 .5 gallons per minute times q2 over 20 or you could just send it out as four gallons per minute it's gonna it's gonna be the same so this one we get three ounces per gap per minute plus one tenth q1 minus 340s q2 minus 2 .5 over 20, so 25 over 200, 5 over 1 eighth, minus 1 eighth q2.

05:21 So 3 plus 1 tenth, q1, that's 540, so that's 8 over 40 minus 1 5th q2.

05:34 Dq1 dt, dq2, dt...

Question

Please give Ace some feedback