Differinal Equations You have a 5 gallon fresh water fish tank, which you have n

Question

Differinal Equations

You have a 5 gallon fresh water fish tank, which you have not cleaned in a while, and the water looks pretty cloudy (it should be clear). You take a sample and determine that there is 50 grams of pollutant per gallon of water. A healthy tank should have less than 10 grams of pollutant per gallon and you decide your goal is to get to 5 g of pollutant per gallon. Initially there is 4 gallons of water in the tank. To not shock your poor fish, you decide to dilute the pollutant by adding 1 gallon of clean water every 30 minutes and drain the mixed water at the same rate. (a) At this rate of putting water in and taking water out at the same rate, how long does it take to reach a pollutant level of 5 g/gal? Be sure to define all variables in a table!

Explanation / Answer

Solution to part (a)

Lets start with the given parameters

Total capacity of the tank is 5 gallons

Initial amount of water present in the tank is 4 gallons

Initially there is 50 grams of pollutant per gallon of water

Therefore, total amount of pollutant prtesent in the water is 50*4 = 200 grams

Our goal to get the amount of pollutant per gallon of water is 5 gramNow to achieve this we are adding 1 gallon of fresh water every 30 minutes and alongwith removing the 1 gallon of mixed water every 30 minutes

Lets take the amount of water present in tank to be W (in gallons) at time equat to T (in minutes)

Also lets take the amount of pollutant per gallon of water to be P (in gram/gallon) and total amount of pollutant in tank to be K (in grams)

Therefore all the variables and their values according to the time is listed below in tabular form

Therefore from the above table we can see that the final concentration of the pollutant in the tank is 3.2768 gram/gallon which is less that 5 gram/gallon

Now if it is required to make the final amount of the water in the tank to be equal to the amount of water present in the tank initially then the time required to reach a pollutant level of 5g/gal is 180 minutes which is equal to 3 hours

and if it not requires to make the final amount of water to be equal to initial one then the time required to reach a pollutant level of 5g/gal is 150 minutes which is equal to 2 hours and 30 minutes.

Solution to part (b)

Now after 1 hour the rate at which water is being drained gets reduced to 1 gallon every hour from 1 gallon every 30 minutes. That means the water will be drained at the rate of 0.5 gallons every 30 minutes. According to this our defined variables from part (a) will be changed to following values

Let G be the amount of water on the floor at time T

(i) Tank will be full after 1 hour when only 0.5 gallons of water will be drained and 1 gallon of water will be added and will be full after every 30 minutes after initial 1 hour.

After 4 hours total of 3.5 gallons of water will be on the floor.

(ii) The mass of the pollutant in the tank when tank first fills is 144 grams.

(iii) Now after 1hour the rate of inlet water is also reduced to 0.5 gallons per 30 minutes. That means there wont be any water on the floor.

From the above table we can see that the amount of pollutant in the tank after 4 hours is 15.3055008 grams/gallon

which is above the maximum limit of the pollutant in tank i.e., 10 grams/gallon

T (min) W (gallon) K (gram) P (gram/gallon) Stage 1 0 4 200 50 Initial 2 0 5 200 40 Added 1gallon water 3 30 4 160 40 drained 1 gallon water 4 30 5 160 32 Added 1gallon water 5 60 4 128 32 drained 1 gallon water 6 60 5 128 25.6 Added 1gallon water 7 90 4 102.4 25.6 drained 1 gallon water 8 90 5 102.4 20.48 Added 1gallon water 9 120 4 81.92 20.48 drained 1 gallon water 10 120 5 81.92 16.384 Added 1gallon water 11 150 4 65.536 16.384 drained 1 gallon water 12 150 5 65.536 3.2768 Added 1gallon water 13 180 4 62.2592 3.2768 drained 1 gallon water

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