Spring ’01 Math 238 Exam 1 Review Prof. Brick
1. A tank with a capacity of 1000 gal initially contains 100 gal of water with 50 lb of salt
dissolved in it. Water containing 2 lb of salt per gal flows into the top of the tank at a rate
of 4 gal/min. A well-stirred solution flows out of the bottom of the tank, but at a rate of
3 gal/min. How much salt is in the tank before it starts to overflow ?
2. To celebrate getting an “A” in Math 238, you go out drinking. But you very foolishly
decide to drive (and you should be ashamed of yourself). You get arrested at 10pm. At
11pm, your blood alcohol is measured and found to be 0.08%. An hour later, it is measured
again and is found to be 0.05%. If the legal limit for alcohol content in the blood is 0.10%,
were you legally drunk when you were arrested ?
3. The half-life of C14 is 5568 years. How old is a fossil that contains 10% of the C14 found
in living things ?
4. Solve the following DE’s:
(a) dy
dt +2ty =t(b) dy
dt +1
ty= sin(t), t>0
5. Solve the following IVP’s:
(a) tdy
dt −3ty =te4t,y(0) = 1 (b) dy
dt −y=t, y(0)=2
6. Solve the following equations, obtaining an explicit solution if possible:
(a) (y2(1 + x2)) dy
dx = arctan(x) (b) dy
dx =xex−1
yex+e(y+x)
7. Without solving them, use a slope field approach to sketch the solutions to the following
differential equation. Indicate those solutions that are equilibriums. You should be able
to do this without your calculator.
(a) dy
dt =−y(y−5) (b) dy
dt =−(y−1)2(y−4)
8. Using a two compartment model, give a linear cascade system of DE’s that describes
the effect of taking a cold pill if the observed data shows an absorption half-life of one hour
and a clearance half-life (after maximum absorption) of eight hours. Solve the system of
equations obtained.
9. Find the first three Picard iterates for the IVP y=x−y2with y(0)=2.
