A tank originally contains 5 lb of salt dissolved in 200 gal of water: Starting at time t=0, a salt solution containing 0.10 lb/gal is to be pumped into the tank at a constant rate and the well-stirred mixture is to flow out of the tank at the same rate.

a)The pumping is to be done so that the tank contains 15 lb of salt after 20 min of pumping. At what rate must the pumping occur in order to achieve this objective?
b)Suppose the objective is to have 25 lb of salt in the tank after 20 min. Is it possible to achieve this objective?

To solve this problem, we'll use the concept of rates and the equation:

Rate in - Rate out = Rate of change

Let's start with part a)

a) The pumping is to be done so that the tank contains 15 lb of salt after 20 minutes of pumping.

We already know that the initial amount of salt in the tank is 5 lb, and it remains constant throughout the process.

Let's assume the rate at which the salt solution is pumped into the tank is 'x' lb/gal/min.

The rate at which salt enters the tank is then: x lb/gal/min * (rate of pumping) gal/min

The rate at which salt leaves the tank is: (5 lb/200 gal) * (rate of pumping) gal/min

Now, we can set up our equation:

Rate in - Rate out = Rate of change

x * (rate of pumping) - (5/200) * (rate of pumping) = (15-5)/20

Simplifying the equation:

(x - 5/200) * (rate of pumping) = 10/20

(rate of pumping) = (10/20) / (x - 5/200)

(rate of pumping) = 20 / (2 * (x - 5/200))

(rate of pumping) = 10 / (x - 5/200)

Next, let's move on to part b)

b) Suppose the objective is to have 25 lb of salt in the tank after 20 minutes. Is it possible to achieve this objective?

Using the same equation as before, we can substitute 25 lb for the desired amount of salt:

(rate of pumping) = 20 / (2 * (x - 5/200))

If we plug in 25 for the desired amount of salt, the equation becomes:

(rate of pumping) = 20 / (2 * (x - 5/200))
(rate of pumping) = 20 / (2 * (x - 5/200))

We can see that there is no possible value for (rate of pumping) that would result in a real solution for x. Therefore, it is not possible to achieve the objective of having 25 lb of salt in the tank after 20 minutes.