Rcantorjr20ur1258idn
Answered

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A tank with a capacity of 500 gal originally contains 200 gal of water with 100 lb of salt
in solution. Water containing 1 lb of salt per gallon is entering at a rate of 3 gal/min, and
the mixture is allowed to flow out of the tank at a rate of 2 gal/min. Find the amount
of salt in the tank at any time prior to the instant when the solution begins to overflow.
Find the concentration (in pounds per gallon) of salt in the tank when it is on the point of
overflowing. Compare this concentration with the theoretical limiting concentration if the
tank had infinite capacity.

Sagot :

Rayadejucosidn

Answer:

200 + ( 3 - 2 ) t = 200 + t, y(0) = 100

y (t) = 3 - 2y (t) , y (0) = 100

       200 + t

y ' t +       2        y = 3, y(0) = 100

        200 + t

h = ∫ P dt = 2 ∫     dt         = ㏑ ( t + 200)^2

                        200 + t

e^h = e^㏑ (200+t)^2 = (200 + t)^2

y(t) = e^-h (c+ ∫ Qe^h dt )

=         1              ( c + 3 (200 + t )^2  dt )

  ( 200 + t )^2

=       c         + 200 + t

 ( 200 + t )^2

100 = y(0) =         c              + 200 + 0 ⇒ - 100 =  c

                     ( 200 + t ) ^2                                   200^2

                                                             ⇒ c = - 4000000 = - 4.10^6

y(t) = -       4.10^6              + 200 + t

             ( 200 + t )^2

500 - 200 = 300 gal.          Concentration at t = 300 =     484         = 0.968

                                                                                              500

y ( 300 ) = -    4.10^6        + 200 + 300 + - 16 + 500 + 484

                    ( 200 + t )^2

lim       (        y(t)             ) ^2  =.       lim       3.10^6             +   ( 200 + t )

t⇒∞        200 + 300                      t⇒∞.      ( 200 + t )^2.                        

                                                                             200 + t

                                               = lim         (       4.10^6           + 1)

                                                t ⇒ ∞           ( 200 + t )^3

                                                = 1