First name:   Last name:

1.  A tank originally contains 100 L of fresh water. Then water containing 50 g/L of salt is pumped into the tank at 4 L/min, and the saline solution leaves the tank at the same rate. At all times, the solution is well-mixed. Find the concentration of salt in the tank after 20 min.

b.  15.2 g/L

b.  19.8 g/L

c.  23.4 g/L

d.  27.5 g/L

e.  31.5 g/L

2.  When will the concentration of the salt in the tank (from question 1) reach 35 g/L?

b.  10 min

b.  15 min

c.  25 min

d.  30 min

e.  35 min

3.  When will the concentration of the salt in the tank (from question 1) reach 55 g/L?

b.  15 min

b.  30 min

c.  45 min

d.  60 min

e.  The concentration will never be 55 g/L.

4.  Consider the following initial value problem: \(\frac{dy}{dx} + \frac{y}{x} = \frac{1}{x-2}\) such that \(y(1) = \frac{1}{2}\). Then we can state that the initial value problem definitely has a solution

b.  for any real x.

b.  for \( 0 < x < 2\).

c.  for \( \frac{1}{2} < x < 3\).

d.  for \( 1 < x < 4\).

e.  for \( 1 \leq x \leq 4\).

5.  Consider the following initial value problem: \(\frac{dy}{dx} + \frac{y^2}{x} = \frac{1}{x-2}\) such that \(y(1) = \frac{1}{2}\). Then we can state that the initial value problem definitely has a solution

b.  for any real x.

b.  for \( 0 < x < 2\).

c.  for \( \frac{1}{4} < x < 9\).

d.  for some open interval centered at \(x = 0\).

e.  for some open interval centered at \(x = 1\).

6.   Write out and submit your solution for questions 1-3.