A tank is filled with 100 gallons of water and initially has

A tank is filled with 100 gallons of water and initially has Q0 pounds of sugar mixed into the solution. One gallon per minute of sugar water with a concentration of 0.5 pounds per gallon are pumped into the tank and disperse evenly. One gallon per minute is also removed from the tank. Write a differential equation for Q(t) = { the # of pounds of sugar in the tank at time t minutes.} for the above situation.

Solution

change in Q(t)=rate at which water enters the tank-rate at which it leaves the tank dQ/dt=(0.5 lb/gallon)-(Q/100) 100*dQ/dt=50-Q so 100*dQ/(50-Q)=dt let 50-Q=p dQ=-dp so substituting, 100*(-dp)/p=dt so 100*ln(p)=-t+c c=a constant or p=exp(-0.01*t)*K k= a constant now 50-Q=p sso Q(t)=50-p=50-exp(-0.01*t)*K at t=0,Q(0)=Q0 so Q0=50-K K=50-Q0 hence complete solution:- Q(t)=50-exp(-0.01*t)*(50-Q0)