A 50-gallon tank initially contains 10 gallons of fresh...

Name: A 50-gallon tank initially contains 10 gallons of fresh water. At t = 0, a brine solution containing 2 pounds of salt per gallon is poured into the tank at a rate of 5 gal/min. The well-stirred mixture drains from the tank at a rate of 3 gal/min. Find the amount of salt in the tank at the moment of overflow.
Uploaded: 2021-10-23T03:20:13-08:00
Duration: 9 min 6 s
Channel: Linda Hand
Description: A 50-gallon tank initially contains 10 gallons of fresh water. At t = 0, a brine solution containing 2 pounds of salt per gallon is poured into the tank at a rate of 5 gal/min. The well-stirred mixture drains from the tank at a rate of 3 gal/min. Find the amount of salt in the tank at the moment of overflow.

A 50-gallon tank initially contains 10 gallons of fresh water. At t = 0, a brine solution containing 2 pounds of salt per gallon is poured into the tank at a rate of 5 gal/min. The well-stirred mixture drains from the tank at a rate of 3 gal/min. Find the amount of salt in the tank at the moment of overflow.

Transcript

00:01 This is a problem we have a 50 gallon tank, but it only has 10 gallons of clean water in it.

00:07 So the water's concentration is zero pounds of salt per gallon because it has no salt in it.

00:16 Okay, but this green stuff coming in, coming in at a rate of five gallons per minute, but it has two pounds per gallon of salt in it.

00:27 And then it's getting stirred up in here and it's coming out.

00:31 And when it comes out, it's coming out at a rate of three gallons per minute.

00:37 But now what's its concentration? okay, because we've mixed this two pound per gallon stuff in with the zero pounds per gallon stuff.

00:46 Okay, so first we have to let y equal the number of pounds of salt in the tank at time t.

01:04 So the concentration here will be t over, now not e over, y over because it's y pounds.

01:15 Y pounds.

01:17 Now how many gallons of water are in there? well, there was 10 gallons to start with.

01:23 It's coming in at a rate at 5 gallons per minute, and it's leaving at a rate of 3 gallons per minute.

01:30 So every minute is gaining 2 gallons.

01:33 So there's 10 gallons plus 2 gallons times t minutes.

01:40 So y pounds over.

01:43 10 plus 2t gallons.

01:46 Again, the 2t came from, it's coming in at 5 gallons per minute, 14 minutes minus, it's leaving at a rate of 3 gallons per minute, 14 minutes.

02:08 So it's 5t minus 3t, 2t gallons gained every minute.

02:17 All right, so now we got that all figured out.

02:20 So now we want to know the rate of change, rate of change of y with respect to time equals rate of change of salt in minus rate of change of salt out.

03:00 All right.

03:02 So rate of change of y with respect to time, d, y, d, d, d, t, oops, equals rate of change of salt in.

03:10 It's coming in at 5 gallons per minute with a concentration of 2 pounds per gallon.

03:19 So it's changing at a rate of 10 pounds for minutes.

03:27 Minus the rate of change of salt water out.

03:30 I forgot already.

03:31 It's going out at 3 gallons per minute times y pounds over 10 plus 2t gallons.

03:39 3 pounds per minute.

03:43 Now 3 gallons per minute.

03:49 Times y pounds over 10 plus 2 t gallons.

03:55 So 3y over 10 plus 2 t pounds per minute is the rate of change of the salt...

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