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**A tank contains 300 gallons of water and 100 gallons of pollutants. Fresh water is pumped into the tank at the rate of 2 gal/min, and the well-stirred mixture leaves the tank at the same rate. How long does it take for the concentration of pollutants in the tank to decrease to 1/10 of its original value**

**This problem is really confusing me as it is water that is being pumped in instead of the usual pollutant/waste.**

this is my attempt (i just reversed what i usually do, but this is wrong..)

w(t) : amount of water in tank at time t

w'(t) : rate of change of water in tank at time t

water enters at : 2

water leaves at : 2 x w(t)/100 (i think this is wrong... i actually worked this problem out with /100 and /300, and both are wrong as i dont get a positive answer.)

w'(t) = 2 - w(t)/50

w'(t) + 1/50 w(t) = 2

a(t) = 1/50, b(t) = 2

u(t) = exp (integ(1/50)dt)

u(t) = e^(1/50)t

d/dt (e^(1/50)t w(t) ) = e^(1/50)t x 2

e^(1/50)t w(t) = integ(2e^(1/50)t)

e^(1/50)t w(t) = 2(1/50) e^(1/50)t + C

w(t) = [1/25 e^(1/50)t + C] / e^(1/50)t

sub w(0) = 300

w(0) = [1/25 e^(1/50)0 + C] / e^(1/50)0

300 = [1/25 (1) + C] / 1

C = 7499/25

so w(t) = 1/25 + (7499/25)e^(-1/50)t

w(t) = 1/25 [ 1+ 7499e^(-1/50)t]

c(t) = w(t)/100 (again, is this correct?)

c(t) = 1/25000 [1+7499e^(-1/50)t]

1/10 of original value of pollutants is 10.

10 = 1/25000 [1+7499e^(-1/50)t]

250000 = 1+7499e^(-1/50)t

249999/7499 = e^(-1/50)t

ln (249999/7499) = -1/50t

t= -50 ln (249999/7499)

what did i do wrong. i think i did this whole problem wrong actually, any help would be great, thanks.

this is my attempt (i just reversed what i usually do, but this is wrong..)

w(t) : amount of water in tank at time t

w'(t) : rate of change of water in tank at time t

water enters at : 2

water leaves at : 2 x w(t)/100 (i think this is wrong... i actually worked this problem out with /100 and /300, and both are wrong as i dont get a positive answer.)

w'(t) = 2 - w(t)/50

w'(t) + 1/50 w(t) = 2

a(t) = 1/50, b(t) = 2

u(t) = exp (integ(1/50)dt)

u(t) = e^(1/50)t

d/dt (e^(1/50)t w(t) ) = e^(1/50)t x 2

e^(1/50)t w(t) = integ(2e^(1/50)t)

e^(1/50)t w(t) = 2(1/50) e^(1/50)t + C

w(t) = [1/25 e^(1/50)t + C] / e^(1/50)t

sub w(0) = 300

w(0) = [1/25 e^(1/50)0 + C] / e^(1/50)0

300 = [1/25 (1) + C] / 1

C = 7499/25

so w(t) = 1/25 + (7499/25)e^(-1/50)t

w(t) = 1/25 [ 1+ 7499e^(-1/50)t]

c(t) = w(t)/100 (again, is this correct?)

c(t) = 1/25000 [1+7499e^(-1/50)t]

1/10 of original value of pollutants is 10.

10 = 1/25000 [1+7499e^(-1/50)t]

250000 = 1+7499e^(-1/50)t

249999/7499 = e^(-1/50)t

ln (249999/7499) = -1/50t

t= -50 ln (249999/7499)

what did i do wrong. i think i did this whole problem wrong actually, any help would be great, thanks.