A tank with capacity of 700 gal of water originally contains 300 gal of water wi

ID: 821809 • Letter: A

Question

A tank with capacity of 700 gal of water originally contains 300 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.

Let Q(t) lb. be the

amount of salt in the tank, V(t) gal be the volume of water in the tank.

1) Find Q(t), the amount of salt in the tank at any time prior to the instant when the solution begins to overflow.

2) Find the concentration (in pounds per gallon) of salt in the tank when it is on the point of overflowing.

Round your answer to three decimal places.

3) Find the theoretical limiting concentration if the tank had infinite capacity.

v(t)= 300+t

t=300min

dA/dt = 3-2(A/300+t)

A'+2(A/300+t)= 3

then intergal, we get

u= (300+t)^2

multiply these through

[ (300+t)^2*A]' = (300+t)^2*A' +2 (300+t)A = 3(300+t)^2

intergal bothsides we get

(300 + t)2A = (300 + t)3 + C1

A = (300 + t) + C1(300 + t)^-2:

Using A(0) = 100, we have

100 = 300 + C1(300)^-2

C1 = -100(300)^2:

Thus,

A(t) = (300 + t) - 100(300)^2(300 + t)^-2:

t = 300 there are

A(300) = 600 -100(300)^2(600)^-2

pounds of salt in the tank. The volume at this time is 500 gallons. Thus, the concentration

[600 -100(300)^2(600)^-2] /700

= 0.821 lbs/gal:

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