Respuesta :
Answer:
the amount of time until 23 pounds of salt remain in the tank is 0.088 minutes.
Step-by-step explanation:
The variation of the concentration of salt can be expressed as:
[tex]\frac{dC}{dt}=Ci*Qi-Co*Qo[/tex]
being
C1: the concentration of salt in the inflow
Qi: the flow entering the tank
C2: the concentration leaving the tank (the same concentration that is in every part of the tank at that moment)
Qo: the flow going out of the tank.
With no salt in the inflow (C1=0), the equation can be reduced to
[tex]\frac{dC}{dt}=-Co*Qo[/tex]
Rearranging the equation, it becomes
[tex]\frac{dC}{C}=-Qo*dt[/tex]
Integrating both sides
[tex]\int\frac{dC}{C}=\int-Qo*dt\\ln(\abs{C})+x1=-Qo*t+x2\\ln(\abs{C})=-Qo*t+x\\C=exp^{-Qo*t+x}[/tex]
It is known that the concentration at t=0 is 30 pounds in 60 gallons, so C(0) is 0.5 pounds/gallon.
[tex]C(0)=exp^{-Qo*0+x}=0.5\\exp^{x} =0.5\\x=ln(0.5)=-0.693\\[/tex]
The final equation for the concentration of salt at any given time is
[tex]C=exp^{-3*t-0.693}[/tex]
To answer how long it will be until there are 23 pounds of salt in the tank, we can use the last equation:
[tex]C=exp^{-3*t-0.693}\\(23/60)=exp^{-3*t-0.693}\\ln(23/60)=-3*t-0.693\\t=-\frac{ln(23/60)+0.693}{3}=-\frac{-0.959+0.693}{3}= -\frac{-0.266}{3}=0.088[/tex]
