1. A tank initially contains 100 gallons of pure water. Water with concentration

Question

1. A tank initially contains 100 gallons of pure water. Water with concentration of 3 ounces of salt per gallon in to the tank at the rate of 5 gallons per minute. The mixed solution flows out at a rate of 3 gallons perminute. Find the amount of salt in the tank when it contains 200 gallons?

2. Pure water flows into a tank at a rate of 100 gallons per minute. the tank initially contains 100 gallons of salt with a concentration of 5 onces per gallon. The mixed fluid flows out of the tank at a rate of 100 gallons per minute. how long will it take for the salt concentation in the tank to reach 1 ounce per gallon?

Explanation / Answer

1.

Let S(t) be he amount of salt in the tank at time t

Rate of salt flowing into tank

3 ounces of sat/gallon * 5 gallons/min = 15 ounces/min

Rate of salt flowing out of the tank :

S ounces / 200 gallon * 3 gallons/min = 3S/200 ounces/min

now dS/dt = rate of salt flowing in - rate of salt flowing out

dS/dt = 15 - 3S/200

200dS/(3000 - 3S) =dt

integrate both sides

=> 200 *[ln(3000 - 3S)]/(-3) =t + C

-200/3 * [ln(3000 - 3S)] = t + C

[ln(3000 - 3S)] = -3t/200 + D

3000 - 3S = e^[(-3t/200) + D]

3S = 3000 - e^[(-3t/200) - D]

=> S(t) = 1000 - 1/3*e^[-3t/200]*e^(-D)

S(t) = 1000 - E*e^[-3t/200] ---------> This is the amount of salt in the tank when it contains 200 gallons

E is the integration constant

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