Solve the problem.
-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a solution containing 2 lb/gal of concentrate enters the tank at the rate of 4 gal/min, and the well-stirred mixture is withdrawn at the rate of 3 gal/min.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?
A) 187.5 pounds
B) 200 pounds
C) 150 pounds
D) 175 pounds

Question

Quizplus · Accepted Answer

The tank starts with 50 gallons of distilled water. The solution entering the tank has a concentration of 2 lb/gal. The tank is filled when an additional 50 gallons have been added, since it starts half full (100 gal capacity). The solution is added at 4 gal/min, but the tank is drained at 3 gal/min, effectively adding 1 gal/min of the solution to the tank. It will take 50 minutes to fill the tank. Since each gallon of incoming solution contains 2 lbs of concentrate, and we're adding 4 gallons per minute, the tank receives 8 lbs of concentrate per minute. However, considering the net addition of 1 gal/min to the tank, we effectively add 2 lbs of concentrate per minute over 50 minutes. Thus, the tank will contain $50 \times 2 = 100$ lbs of concentrate from the incoming solution. Since the tank initially contained distilled water (no concentrate), the total amount of concentrate when the tank is full will be 100 lbs. However, this calculation overlooks the fact that the concentration of the solution entering the tank and the volume rate at which it enters should directly determine the final amount of concentrate when the tank is full, without subtracting the outflow because the question asks for the total amount when the tank is full, not the net increase per minute. The correct approach involves calculating the total volume of concentrate added until the tank is full, considering the concentration and rate of inflow. Given the mistake in the initial explanation, let's correct the approach:The correct calculation involves the total amount of concentrate added to the tank from the incoming solution over the time it takes to fill the tank. The inflow rate is 4 gal/min with a concentrate of 2 lb/gal. The tank reaches full capacity when the volume difference (inflow - outflow) fills it up. However, the key is to focus on the concentrate added:Since the solution enters at 4 gal/min and contains 2 lb/gal of concentrate, for every minute, 8 lbs of concentrate are added to the tank. The question essentially simplifies to how much concentrate is added until the tank is full, disregarding the outflow because it doesn't affect the total amount of concentrate added, just the time it takes to fill the tank. Given the misunderstanding in the initial explanation, the correct answer should reflect the total pounds of concentrate when the tank is full, based on the inflow concentration and rate, and the capacity of the tank. The calculation provided initially doesn't correctly account for these factors in determining the final amount of concentrate. The correct approach should calculate the total concentrate based on the inflow rate and concentration until the tank reaches its full capacity, considering the initial condition and how the inflow contributes to the concentrate level.

Solve the Problem. -A 100 Gal Tank Is Half Full of Distilled Water