Compartmental Analysis Word Problem

[pwood]

Hi, I am asking for some help with the following problem:

Beginning at time t=0, fresh water is pumped atthe rate of 3 gal/min into a 60-gal tank initially filled
with brine. The resulting less-and-less salty mixture
overflows at the same rate into a second 60-gal tank
that initially contained only pure water, and from
there it eventually spills onto the ground. Assuming
perfect mixing in both tanks, when will the water in
the second tank taste saltiest? And exactly how salty
will it then be, compared with the original brine?

My approach was to ask myself what the input brine amount was, so I concluded that input = 0.
My initial equation is:
ds/dt = 0 - s/60

Could someone tell me if this is correct and what my next step would be, in regards to the second equation? I know how to solve the differential equation, but I'm not sure about the next step. Thanks in advance.

 

[HallsofIvy]

It's impossible to be sure what you are doing because you are not defining your terms. You have two different tanks. What does "s" represent? If you mean s to represent the amount of salt in the first tank, in, say, pounds, then your equation is almost right. Because there are 60 gallons of water in the tank, the amount of salt in each gallon is s/60 pounds per gallon but the water is leaving the tank at 3 gal/min so the salt is leaving at 3(s/60)= s/20 pounds per min. The differential equation should be ds/dt= -s/20.

Now, let r be the amount of salt, in pounds, in the second tank. All of the salt coming from the first tank, s/20 pounds per min, comes into the second tank while the mixture in that tank, r/60 pounds per gal, leaves at 3 gal/min so the salt is leaving at 3(r/60)= r/20 pounds per min. The equation governing the amount of salt in the second tank is dr/dt= s/20- r/20.

One question asked is "Assuming perfect mixing in both tanks, when will the water in the second tank taste saltiest?" That is, for when will r(t) be maximum?

The other is "And exactly how salty will it then be, compared with the original brine?" So you want to find r(t)/s(0).