a) A very large tank initially contains 15 gallons of saltwater containing 6 pou

Question

a) A very large tank initially contains 15 gallons of saltwater containing 6 pounds of salt. Saltwater containing 1 pound of salt per gallon is pumped into the top of the tank at a rate of 2 gallons per minute, while a well-mixed solution leaves the bottom of the tank at a rate of 1 gallon per minute. Right and solve a rate of change differential equation.

b) What does your solution predict about the amount of salt in the tank in the long run? How about the concentration?

c) Explain how you can make sense of the predictions from by using the differential equation itself.

Explanation / Answer

a)intial salt=>s(0)=6 pounds

rate in=1*2=2pound per minute

rate out=(s(t)/15)*1=s(t)/15 where s(t) is amount of salt at any time 't'

ds/dt=2-s(t)/15

ds/30-s(t)=dt/15

on integtating

ln(s(t)-30)=-t/15+C C is a constant

s(t)-30=Ce-t/15

s(t)=30+Ce-t/15

but s(0)=6

6=30+C=>C=-24

s(t)=30-24e-t/15

b)in long run t=infinity at t=infinity =>e-t/15=0

hence s(infinity)=30pounds

c)it provides a significant or appropriate estimations for concentration of salt solution depending on rate of inflow and out flow

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