Question

A tank initially contains 400 gallons of fresh water. At time t=0, a brine solution with a concentration of 0.1 pounds of salt per gallon enters the tank at a rate of 1 gallon per minute, and the well-stirred mixture flows out at a rate of 2 gallons per minute. a. Solve the differential equation to find an expression for the pounds of salt, x, at time t. b. What is the concentration of salt after 60 minutes? c. When will the concentration of salt reach 0.05 pounds per gallon? d. What is the concentration of salt when the tank contains 100 gallons of brine?

          A tank initially contains 400 gallons of fresh water. At time t=0, a brine solution with a concentration of 0.1 pounds of salt per gallon enters the tank at a rate of 1 gallon per minute, and the well-stirred mixture flows out at a rate of 2 gallons per minute.

a. Solve the differential equation to find an expression for the pounds of salt, x, at time t.
b. What is the concentration of salt after 60 minutes?
c. When will the concentration of salt reach 0.05 pounds per gallon?
d. What is the concentration of salt when the tank contains 100 gallons of brine?

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Added by Sofia P.

Differential Equations and Linear Algebra

Stephen W. Goode, Scott A. Annin 4th Edition

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Solved by Expert Sam Stansfield

03/31/2023

Step 1

Let x(t) be the amount of salt in the tank at time t. The rate of salt entering the tank is 0.1 pounds per gallon * 1 gallon per minute = 0.1 pounds per minute. The rate of salt leaving the tank is (x(t) pounds / (400 - t) gallons) * 2 gallons per minute. So, the Show more…

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A tank initially contains 400 gallons of fresh water. At time t=0, a brine solution with a concentration of 0.1 pounds of salt per gallon enters the tank at a rate of 1 gallon per minute, and the well-stirred mixture flows out at a rate of 2 gallons per minute. a. Solve the differential equation to find an expression for the pounds of salt, x, at time t. b. What is the concentration of salt after 60 minutes? c. When will the concentration of salt reach 0.05 pounds per gallon? d. What is the concentration of salt when the tank contains 100 gallons of brine?

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Transcript

00:01 We have a tank that starts out with 400 gallons of fluid in it with an initial concentration of zero.

00:08 We're going to bring stuff in it.

00:10 It has 0 .1 pounds per gallon of salt at a rate of one gallon per minute.

00:15 We're going to pump it out at a rate of two gallons per minute.

00:19 X is concentration.

00:20 A is amount.

00:25 So our standard differential equation is d -a -d -t is the rate of stuff coming in times the concentration minus the rate of stuff going out times the concentration going out.

00:37 And x out is a of t over v of t.

00:46 So there's our basic differential equation.

00:49 The volume is v0 plus r in minus r out times t, which comes out to be 400 minus t.

01:00 Or t is in minutes.

01:04 And then we can plug in our volume and our other numbers.

01:08 So you get 2a over 400 minus t.

01:12 Rn times x in.

01:14 Is 0 .1.

01:28 So we look at d by dt, 400 minus t to the minus 2 times a.

01:34 That's our integrating factor, actually.

01:37 We can take that derivative and we end up with minus t to the minus two times the thing on the left hand side of the differential equation, which is just equal to 0 .1.

02:21 So the thing up at the top of all this, equalities is a perfect derivative...

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