A tank contains 1000 l of pure water brine that contains 005...

A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15 L/min. (a) How much salt is in the tank after t minutes? (b) How much salt is in the tank after 50 minutes? (Round the answer to one decimal place.)

Transcript

00:01 Okay, this problem starts with a thousand liters of clear water in the tank.

00:06 Its concentration is zero pounds of salt per, this must be per liter, per liter because it's plain water.

00:17 And then there's two pipes with stuff flowing in, and the green stuff is coming in at a rate of five liters per minute, and its concentration is 0 .05 kilograms per liter of salt.

00:32 And then the black one is coming in at 10 liters per minute, and its concentration is 0 .04 kilograms per liter.

00:41 Okay, then that's getting mixed up, and then it's coming out, and it's coming out at a rate of 15 liters per minute, but we don't know what the concentration is.

00:51 We have to figure it out so that we can answer the question.

00:56 Okay, so what we need to do first is let y equal the number of, of kilograms of salt in the tank at time tea.

01:19 All right, so the concentration will be y kilograms over.

01:27 All right, so how much gallons are in there at any one time? well, there's every minute, five liters of the green stuff and 10 liters of the black stuff, gets in there, but every minute 15 liters of the mixed up stuff comes out.

01:48 So in fact, in this one, we just get a thousand here.

01:52 There's no change.

01:54 As you put it in and mix it up, the same amount is coming out as going in.

01:59 So there is always a thousand liters in there.

02:04 Okay, so now we want to find the rate of change of y with respect to time.

02:16 And that's equal to the change, wait, it's equal to the, not the change, the rate, the rate of y in, minus the rate of y out.

02:44 Okay, so the rate of change of y with respect to time, that's d, y, d, t.

02:50 Rate of change of y in.

02:51 Well, the green one is coming in at five liters per minute, and it has 0 .0 .4.

02:58 Kilograms per liter so 0 .05 kilograms per minute so 0 .05 of the green one plus also coming in i know five times 0 .05 so 0 .25 also coming in we have the black one and it's coming in at 10 liters per minute at a concentration of 0 .04 kilograms per liter so 0 .4 kilograms per minute minus the rate out 15 liters per minute oops liters per minute times y kilograms per liter so why kilograms per thousand liters so 15 over 1 ,000 y so d t equals 0 .65 minus 10 and 100 with 0 .015y...

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