A tank with capacity of 600 gal of water originally contains 300 gal of water wi

Question

A tank with capacity of 600 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. Find the amount of salt in the tank at any time prior to the instant when the solution begins to overflow. Your answer should be a function in terms of t. Q(t) = Find the concentration (in pounds per gallon) of salt in the tank when it is on the point of overflowing. Round your answer to 3 decimal places. Concentration = Find the theoretical limiting concentration if the tank had infinite capacity.

Explanation / Answer

Max capacity=600 gal

initial filled= 300 gal

remaining =300 gal.

initial salt amount is 100 lb.

as 1lb salt /gal water is adding at rate 3gal/min.

overflow starts when 300 gal of water is poured i.e., after 100min.

before that ,let t min

the water poured will be =3t gal

total V(t)=300+3t gal

and,

salt added = 3t lb

so total amount of salt Q(t)=(100+3t) lb

concentration=Q/V=100+3t/300+3t

at t=100 min(overflow starts)

C=100+300/300+300=400/600=2/3 lb/gal

if tank had infinite capacity

Lt x-> Q(t)/V(t)=Lt x-> 100+3t/300+3t=Lt x->(3+100/t)/3+300/t)

                                                          =3+0/3+0=1 lb/gal(limiting concentration)

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