Brine containing 1.5lb/gal of salt enters a tank at the

oliviayychengwh

oliviayychengwh

Answered question

2022-01-03

Brine containing 1.5lb/gal of salt enters a tank at the rate of 2gal/min and the mixture well stirred leaves at the same rate. The tank initially contains 100 gal of brine with10 lbs of dissolved salt. When will the amount of salt in the tank be 20 lbs?

Answer & Explanation

Gerald Lopez

Gerald Lopez

Beginner2022-01-04Added 29 answers

y= amount of salt at time t
ri=2
ci=1.5
ro=2
v(t)=v0+(riro)t
v(t)=100+(22)t
v(t)=100+0t
v(t)=100
co=yv(t)
co=y100
DE is given by
dydt=riciroco
dydt=21.52y100
dydt=3y50
dydt=y50+3
dydt=(y503)
dydt=150(y150)
dydt=0.02(y150)
dy(y150)=0.02dt
dy(y150)=0.02dt
ln(y150)=0.02t+c
y150=e0.02t+c
y150=e0.02tec
y150=e0.02tC
y150=Ce0.02t
Matthew Rodriguez

Matthew Rodriguez

Beginner2022-01-05Added 32 answers

dydt=(rate in)(rate out)
dydt=2(15)(y100)
dydt+y100=3(1)
dydt+p(x)y=q(x)
Which is a first order linear ode.
I.f=epdx=e1100dx=ex100
y×IF=q×IF.dx
y.ex100=3ex100dx
yex100=300ex100+c
y=300+cex100
So y=300+cet100(2)
y(0)=10100=110
(2) 300+ce0=110C=110300
C=2.99910=299.9
So (2) 300299.9et/100=100
299.9et100=200
et100=0.66688
t100=log(0.66688)
t=100log(0.66688)
t=100×0.4051
t=40.51

karton

karton

Expert2022-01-09Added 613 answers

Let x(t)= The amount of salt in the tank at time t.
Rate of increase of x,
dxdt=Rate of salt inRate of salt out
(Rate)in=3lb/gal×4gal/min
=12lb/min(Rate)out=2gal/min×x(t)2t+20lb/gal=2x2t+20lb/mindxdt=122x2t+20dxdt=+2x2t+20=12This is linear first order differential equation of the formdxdt+px=qp=22t+20 q=12Integrating factor (I.F)=epdt=e22t+20dtPut 2t+20=z2dt=dzIF=edzz=eln|z|=eln|2t+20|IF=2t+20The solution isx×I.F=q×IFdtx(2t+20)=12(2t+20)dtx(2t+20)=122(t+10)dt=24(t+10)dt=24[t22+10t]+cx(t)=12t2+240t+c2t+20Given, x(0)=00=12(0)2+240×0+c2×0+200=c20After t=5minx(5)=12(5)2+240×5+20(2×5)+20=(12×25)+1200+2010+20=300+1200+20300=152030=50.67lbc=0x(t)=12t2+240t2t+20=2(6t2+120t)2(t+10)x(t)=6t2+120tt+10After t=5minx(5)=6(5)2+120(5)5+10=(6×25)+60015=150+60015=75015=50lb
The amount of salt in the tank after 5 min is 50lb.

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