Question: A tank initially contains 100 gal of brine in which 70 lb of salt are dissolved. A brine containing 2 lb/gal of salt runs into the tank at the rate of 6 gal/min. The mixture is kept uniform by stirring and flows out of the tank at the rate of 5 gal/min. Let $y$ represent the amount of salt at time $t$. Complete parts a through e.
a. At what rate (pounds per minute) does salt enter the tank at time $t$?
b. What is the volume of brine in the tank at time $t$?
1. **State the problem:**
We have a tank with an initial volume of 100 gallons containing 70 pounds of salt. Brine with 2 pounds of salt per gallon flows in at 6 gallons per minute, and the mixture flows out at 5 gallons per minute. We want to find:
a. The rate at which salt enters the tank at time $t$.
b. The volume of brine in the tank at time $t$.
2. **Part a: Rate salt enters the tank**
- The incoming brine concentration is 2 lb/gal.
- The inflow rate is 6 gal/min.
The rate salt enters the tank is given by:
$$\text{Rate in} = (\text{concentration}) \times (\text{flow rate}) = 2 \times 6 = 12 \text{ lb/min}$$
So, salt enters at a constant rate of 12 pounds per minute.
3. **Part b: Volume of brine in the tank at time $t$**
- Initial volume: 100 gal
- Inflow rate: 6 gal/min
- Outflow rate: 5 gal/min
Net inflow rate = $6 - 5 = 1$ gal/min
Volume at time $t$ is:
$$V(t) = 100 + 1 \times t = 100 + t \text{ gallons}$$
**Final answers:**
a. Salt enters at $12$ lb/min.
b. Volume of brine at time $t$ is $100 + t$ gallons.