A 100 gallon tank initially contains 40 gallons of water with 20 lbs of salt dis

ID: 2985408 • Letter: A

Question

A 100 gallon tank initially contains 40 gallons of water with 20 lbs of salt dissolved in

it. At time t = 0, brine containing 2 lbs of salt per gallon begins to be pumped into

the tank at a rate of 3 gallons per minute. Also at t = 0, a drain is opened at the

bottom of the tank, that lets out 1 gallon of well-mixed brine per minute.

(a) Find an explicit expression for the amount of salt in the tank after t minutes, and

the amount of salt in the tank at the moment it is completely full.

(b) Now, consider the same situation, except that brine entering the tank has a vary-

ing concentration of salt, beginning at 2 lbs per gallon at t = 0 and increasing

linearly to 4 lbs per gallon over the rst 30 minutes. Set up, but do not solve, a

dierential equation for the amount of salt in the tank at time t:

DO NOT SKIP/ omitting steps , Please explain what difrential/ integration rule was used and highlight it and EXPLAIN All steps

salt at time t incoming = 3gallons /min x 2 lbs/gallon x t in min = 3 x 2 x t lbs/min

salt initially = 20 lbs

outflow = 1 x final concentration of salt x t

total water at time t = 3 x t + 40 - 1t gallons = 40+ 2t gallons

concentration of salt at time t = [20 + 3t ] lbs / [ 40 + 2t ] gallons

so outflow = 1 x [20 + 3t ] lbs / [ 40 + 2t ] x t

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