Question
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.
Answers
Bot
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.
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.