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A 400-gallon tank initially contains 200 gallons of water

Chapter , Problem 25

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QUESTION:

A 400-gallon tank initially contains 200 gallons of water containing 2 parts per billion by weight of dioxin, an extremely potent carcinogen. Suppose water containing 5 parts per billion of dioxin flows into the top of the tank at a rate of 4 gallons per minute. The water in the tank is kept well mixed, and 2 gallons per minute are removed from the bottom of the tank. How much dioxin is in the tank when the tank is full?

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QUESTION:

A 400-gallon tank initially contains 200 gallons of water containing 2 parts per billion by weight of dioxin, an extremely potent carcinogen. Suppose water containing 5 parts per billion of dioxin flows into the top of the tank at a rate of 4 gallons per minute. The water in the tank is kept well mixed, and 2 gallons per minute are removed from the bottom of the tank. How much dioxin is in the tank when the tank is full?

ANSWER:

Step 1 of 3

we are told that 5 parts per billion of dioxin flows in at a rate of 4 gallons per minute. So,

and initially it is said that the tank contains 200 gallons of water (which contains 2 parts per billion by weight of dioxin), hence the volume of the water in the tank at time  is given by

and so the concentration of dioxin in the tank is given by

and since the mix leaves the tank at a rate of 2 gallons per minute, then

therefore the differential equation that models the amount of dioxin in the tank is

First we write the equation in the standard form

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