A large tank contains 75 gallons of brine in which 2lb of salt is dissolved. Suppose we pump brine at a concentration of 0.5 pounds per gallon into the tank at a rate of 4 gallons per minute. We then pump the mixture out of the tank at a rate of 3 gallons per minute. How much salt will be in the tank at the moment that there are 100 gallons of fluid in the tank? Round your answer to decimal places.
Let's compute the time it takes for the tank to have 100 gallons of fluid. We will denote this time by $t$. Initially, there are $75$ gallons of fluid. We add brine at a rate of $4$ gal/min and remove it at a rate of $3$ gal/min, so the net increase in fluid each minute is $4-3=1$ gallon/min. Therefore, it takes $t$ minutes to increase the amount of fluid in the tank from $75$ gallons to $100$ gallons, so we solve $75+t=100$ to find $t=25$ minutes.
During these $25$ minutes, we pump brine at a rate of $4$ gal/min, so the total amount of salt we pump into the tank is $0.5\text{ pounds/gal}\times 4\text{ gal/min}\times 25 \text{ min}=50$ pounds.
Note that since the amount of fluid in the tank changed, the concentration of salt in the tank is higher than it was initially. Initially, we had $2$ pounds of salt dissolved in $75$ gallons of fluid, so the initial concentration was $2/75$ pounds/gallon. After pumping brine for $25$ minutes, we have a total amount of $75+4\times 25=175$ gallons of fluid, so the concentration is now the ratio of the total amount of salt to the total amount of fluid, or $50/175=0.285714$ pounds/gallon.
Therefore, when there are $100$ gallons of fluid in the tank, there will be approximately $\boxed{28.57}$ pounds of salt in the tank.
To find how much salt will be in the tank at the moment when there are 100 gallons of fluid in the tank, we need to track the amount of salt added and removed from the tank.
Initially, the tank contains 75 gallons of brine with 2 pounds of salt dissolved.
For every minute, we add brine at a rate of 4 gallons per minute with a concentration of 0.5 pounds per gallon. This means we add 4 * 0.5 = 2 pounds of salt per minute.
At the same time, we remove the mixture at a rate of 3 gallons per minute.
So, for every minute, the net addition of fluid to the tank is 4 - 3 = 1 gallon.
We want to find the moment when there are 100 gallons of fluid in the tank. The initial fluid in the tank is 75 gallons, so there will be 100 - 75 = 25 additional gallons of fluid.
Since there is a net addition of 1 gallon per minute, it will take 25 / 1 = 25 minutes to reach 100 gallons of fluid.
During these 25 minutes, we add 2 * 25 = 50 pounds of salt to the tank.
Therefore, at the moment that there are 100 gallons of fluid in the tank, there will be 50 pounds of salt in the tank.