Problems from an old ma112 exam
(10 pts) 4. A 1000 gallon tank filled with salt water initially contains 120 lbs of salt. A brine solution
(1/4 lb of salt per gallon of water) is pumped into the tank at the rate of 6 gallons per
minute. We may also assume that the well—stirred mixture in the tank is leaving at the same
rate. Let x be the amount of salt in the tank at time t.
(a) Sketch a picture of the tank with the appropriate arrows and rates, and state the IVP
(i.e. the DE and initial condition) which describes x.
(b) Determine x.
(c) Give a reasonable sketch of the solution (let t ≥ 0 and go far enough to see the limiting
behavior).
(10 pts) 5. Suppose that you are at the middle of a 80 m building tossing a 1.2 kg ball upwards with a
speed of 25 m/sec. Assume that gravity and air resistance are the only forces acting on the
ball and that the magnitude of the resistive force is 1.4 times the speed of the ball.
(a) Draw the picture (include axis with zero and direction) and give the IVP for v, the
velocity of the ball.
(b) Solve the differential equation for v.
(c) Give a reasonable sketch of v.
(d) How long does it take for the ball to reach its maximum height?
(10 pts) 1. Let f (x) = x and g(x) = 12 x2 . Give a sketch of the two functions on the same set of axes
and determine the area of the region bounded by the two graphs.
(10 pts) 2. Determine the length of the graph of y = sin(x) for x = 0 to x = π. Set up the integral only.
√
(10 pts) 3. Let R be the region in the first quadrant bounded by the graph of f (x) = x, the y-axis,
and the line y = 2. Find the volume of the solid generated by revolving R about the y-axis
using (note that you need only give a good sketch and set up the integrals)
(a) the disk method
(b) the shell method
√
(10 pts) 4. Suppose that a solid has its base in the first quadrant with boundaries y = x, the x-axis,
and the line x = 4. If the cross sections perpendicular to the y-axis are semi-circles, what is
its volume?
(10 pts) 5. A solid is generated by revolving the region in the first quadrant bounded by y = 1 − x2 ,
the x-axis, and the y-axis about the y-axis.
a) Find the volume of the solid.
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
b) A “cone” shaped region is to be removed from the solid so that the remaining volume is
half the original volume. The “cone” shaped region is formed by rotating the region above
the line y = mx about the y-axis. Find m. (NOTE: this part would not be on a No-Maple
exam, but it would certainly be good study material for the final exam).
y
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x