1. ## Mixture problem

I've been racking my brain and Can't figure part of this question out...

------
A tank initially contains 100 gallons of water with 20 lbs of salt dissolved in int. A solution containing 2 lbs of salt per gallon of water is pumped into the tank at a rate of 1 gallon per minute and the well-mixed solution is pumped out of the tank at a rate of 2 gallons per minute.

A) How much salt will be in the tank after 25 mins?
B) When will there be 55 lbs of salt in the tank? [Note: There are two answers]
C) How much salt will be in the tank after 100 minutes?
------
I started out with my initial equation
dS/dt = rate in - rate out
dS/dt = (2)(1) - (s/100)(2)

This was rewritten into a linear equation form
S' + (1/50)S = 2

and plugged in as
S = (∫e∫1/50 dt * 2 dt) / (e∫1/50 dt)

Some more magic gets me to
S = 100+Ce-t/50

and solving for C using information given in the problem gets me
S=100-80e-t/50

for A its just plug and chug - t=25 and I get S=51.47 lbs.
for B its more plug and chug - S=55 so I get T=28.76 Mins. However I cant find the second answer.
for C its pretty easy - the tank empties at T=100 so it's 0.

I am really stuck on part B. I know there are two answers, because the salt in the tank fills up to a maximum, then the flow of water out begins to overcome the saltwater solution going in, so my salt level in the tank decreases - the whole thing looks like an inverted parabola shape. However my equation doesn't give me this.

2. The water volume isn't fixed. Work on that.
Make sure your rates have the same scale.

3. Yeah I think I figured it out

my initial equation is
dS/dt = 2 - (S/100-t)(2)

so I get
S' + (2/100-t)S = 2

plugging that in I get
S = (∫e2∫1/100-t dt * 2 dt) / (e2∫1/100-t dt)

and integrating I end up with
S = 2(100-t) + C(100-t)2

Plugging in S(0)=20, I get
C= -0.018

so my equation ends up as
S = 2(100-t) - 0.018(100-t)2

A) 48.75 lbs
B) 38.9 mins and 50 mins
C) 0 lbs

I think I figured it out. Double checking with a graphing calculator, the curve looks true and my numbers line up with my T and S at those points.

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•