A 100 gal tank is half full of distilled water. At time = 0, a solution containing 2 lb/gal of concentrate enters the tank at the rate of 4 gal/min, and the well-stirred mixture is withdrawn at the rate of 3 gal/min. When the tank Is full, how many pounds of concentrate will it contain?

Question

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At time = 0, there are 50 gallons of water in the tank. Let's keep track of the amount of concentrate in the tank at every minute.
After 1 minute, 4 gallons of the solution have entered, containing a total of 4 x 2 = 8 pounds of concentrate. Three gallons of the mixture are withdrawn, leaving 51 gallons in the tank. The concentration of the mixture is now:

(50 x 1) + (8 x 1) / 51 = 1.2549 lb/gal

After 2 minutes, another 8 pounds of concentrate have entered, for a total of 16 pounds. Three gallons are withdrawn, leaving 52 gallons in the tank. The concentration of the mixture is:

(50 x 2) + (8 x 2) + (8 x 1) / 52 = 1.4118 lb/gal

After 3 minutes, another 8 pounds of concentrate have entered, for a total of 24 pounds. Three gallons are withdrawn, leaving 53 gallons in the tank. The concentration of the mixture is:

(50 x 3) + (8 x 3) + (8 x 2) / 53 = 1.5377 lb/gal

We can continue this process until the tank is full. We know that the tank will be full after 50 minutes, since it starts at half full and 4 gallons are added per minute. At this point, 200 gallons of solution have entered the tank, containing a total of 400 pounds of concentrate. 150 gallons of the mixture have been withdrawn, leaving 50 gallons of the mixture in the tank. The concentration of the mixture is:

(50 x 50) + (8 x 50) + (8 x 49) + (8 x 48) + ... + (8 x 2) / 100 = 1.875 lb/gal

So the tank contains a total of:

50 gallons x 1.875 lb/gal = 93.75 pounds of concentrate

However, we have to add the 400 pounds of concentrate that entered the tank to get the total amount of concentrate in the tank:

93.75 + 400 = 493.75

We can round this to the nearest half pound, which gives us:
Answer: C) 187.5 pounds

A 100 Gal Tank Is Half Full of Distilled Water