James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 9.3 Separable Equations 607
is kept thoroughly mixed and drains from the tank at the
same rate. How much salt is in the tank (a) after t minutes
and (b) after 20 minutes?
46. The air in a room with volume 180 m 3 contains 0.15% carbon
dioxide initially. Fresher air with only 0.05% carbon dioxide
flows into the room at a rate of 2 m 3 ymin and the mixed air flows
out at the same rate. Find the percentage of carbon dioxide in
the room as a function of time. What happens in the long run?
47. A vat with 500 gallons of beer contains 4% alcohol (by volume).
Beer with 6% alcohol is pumped into the vat at a rate of
5 galymin and the mixture is pumped out at the same rate. What
is the percentage of alcohol after an hour?
48. A tank contains 1000 L of pure water. Brine that contains
0.05 kg of salt per liter of water enters the tank at a rate of
5 Lymin. Brine that contains 0.04 kg of salt per liter of water
enters the tank at a rate of 10 Lymin. The solution is kept
thoroughly mixed and drains from the tank at a rate of
15 Lymin. How much salt is in the tank (a) after t minutes
and (b) after one hour?
49. When a raindrop falls, it increases in size and so its mass at
time t is a function of t, namely, mstd. The rate of growth of the
mass is kmstd for some positive constant k. When we apply
New ton’s Law of Motion to the raindrop, we get smvd9 − tm,
where v is the velocity of the raindrop (directed downward) and
t is the acceleration due to gravity. The terminal velocity of the
raindrop is lim t l ` vstd. Find an expression for the terminal
velocity in terms of t and k.
50. An object of mass m is moving horizontally through a medium
which resists the motion with a force that is a function of the
velocity; that is,
m d 2 s
dt 2
− m dv
dt − f svd
where v − vstd and s − sstd represent the velocity and position
of the object at time t, respectively. For example, think of a
boat moving through the water.
(a) Suppose that the resisting force is proportional to the
velocity, that is, f svd − 2kv, k a positive constant. (This
model is appropriate for small values of v.) Let vs0d − v 0
and ss0d − s 0 be the initial values of v and s. Determine v
and s at any time t. What is the total distance that the object
travels from time t − 0?
(b) For larger values of v a better model is obtained by supposing
that the resisting force is proportional to the square
of the velocity, that is, f svd − 2kv 2 , k . 0. (This model
was first proposed by Newton.) Let v 0 and s 0 be the initial
values of v and s. Determine v and s at any time t. What is
the total distance that the object travels in this case?
51. Allometric growth in biology refers to relationships between
sizes of parts of an organism (skull length and body length, for
instance). If L 1std and L 2std are the sizes of two organs in an
organism of age t, then L 1 and L 2 satisfy an allometric law if
CAS
their specific growth rates are proportional:
1 dL 1
− k 1 dL 2
L 1 dt L 2 dt
where k is a constant.
(a) Use the allometric law to write a differential equation
relating L 1 and L 2 and solve it to express L 1 as a function
of L 2.
(b) In a study of several species of unicellular algae, the
proportionality constant in the allometric law relating
B (cell biomass) and V (cell volume) was found to be
k − 0.0794. Write B as a function of V.
52. A model for tumor growth is given by the Gompertz
equation
dV
− asln b 2 ln VdV
dt
where a and b are positive constants and V is the volume of
the tumor measured in mm 3 .
(a) Find a family of solutions for tumor volume as a function
of time.
(b) Find the solution that has an initial tumor volume of
Vs0d − 1 mm 3 .
53. Let Astd be the area of a tissue culture at time t and let M
be the final area of the tissue when growth is complete.
Most cell divisions occur on the periphery of the tissue
and the number of cells on the periphery is proportional to
sAstd. So a reasonable model for the growth of tissue is
obtained by assuming that the rate of growth of the area is
jointly proportional to sAstd and M 2 Astd.
(a) Formulate a differential equation and use it to show
that the tissue grows fastest when Astd − 1 3 M.
(b) Solve the differential equation to find an expression for
Astd. Use a computer algebra system to perform the
integration.
54. According to Newton’s Law of Universal Gravitation, the
gravitational force on an object of mass m that has been
projected vertically upward from the earth’s surface is
F − mtR 2
sx 1 Rd 2
where x − xstd is the object’s distance above the surface
at time t, R is the earth’s radius, and t is the acceleration
due to gravity. Also, by Newton’s Second Law,
F − ma − msdvydtd and so
m dv
dt − 2 mtR 2
sx 1 Rd 2
(a) Suppose a rocket is fired vertically upward with an
initial velocity v 0. Let h be the maximum height above
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