solve this in general using the integration factor method. then specify your f(x) as given with the initial data.

Let be the amount of brine in the tank at time (we should really write , but that gets annoying)" A tank contains 40 gallons of pure water. Brine with 3 lb of salt per gallon flows in at the rate of 2 gal/min. The thoroughly stirred mixture then flows out at the rate of 3 gal/min.

clearly

now,

note, the concentration out is amount (Q) per volume. the volume is decreasing by 1 gallon every minute, so after t minutes, the volume decreases by t gallons. since we started with 40, the volume after time t is (40 - t)

thus we have

this is a first order ODE, you can solve for

i suppose to mean 20 gallons of the mixture.a. Find the amount of salt in the tank when there are 20 gallons of brine in the tank.

that is just asking for , since we have 20 gallons left after 20 mins. you found above

you want to maximize . that is, solve for and take the local maximum value, this will give you the timeb. When is the amount of salt in the tank greatest? Provide an exact (symbolic) answer as well as an approximation to one decimal place. Use Calculus techniques. "