ASSIGNMENT 2.1-MATHEMATICAL MODELING
Q. No.1 The population of a town grows at a rate proportional to the population present at time t. The
initial population of 500 increases by 15% in 10 years. What will be the population in 30 years? How fast
is the population growing at t = 30?
Q. No. 2 The population of bacteria in a culture grows at a rate proportional to the number of bacteria
present at time t. After 3 hours it is observed that 400 bacteria are present. After 10 hours 2000 bacteria
are present. What was the initial number of bacteria?
Q. No. 3. The rate at which a body cools also depends on its exposed surface area S. If S is a constant,
then a modification of Newton’s Law of cooling is:
dT/ dt= kS(T - Tm), where k< 0 and Tm is a constant. Suppose that two cups A and B are filled with
coffee at the same time. Initially, the temperature of the coffee is 150° F. The exposed surface area of
the coffee in cup B is twice the surface area of the coffee in cup A. After 30 min the temperature of the
coffee in cup A is 100° F. If Tm = 70° F, then what is the temperature of the coffee in cup B after 30 min?
Q. No. 4. A tank contains 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing 1
gram of salt per liter is then pumped into the tank at a rate of 4 L/min; the well-mixed solution is
pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.
Q. No. 5.Two chemicals A and B are combined to form a chemical C. The rate, or velocity, of the
reaction is proportional to the product of the instantaneous amounts of A and B not converted to
chemical C. Initially, there are 40 grams of A and 50 grams of B, and for each gram of B, 2 grams of A is
used. It is observed that 10 grams of C is formed in 5 minutes. How much is formed in 20 minutes? What
is the limiting amount of C after a long time? How much of chemicals A and B remains after a long time?
Q. No. 6. A cantilever beam of length L is embedded at its right end, and a horizontal tensile force of P
pounds is applied to its free left end. When the origin is taken at its free end, as shown in the Figure ,
the deflection y(x) of the beam can be shown to satisfy the differential equation . Find the deflection of
the cantilever beam if w(x) =w0 x, 0<x<L and y(0) = 0, y’(L) =
Q. No. 7 Let us consider the model mx’’+ kx = 0 for simple harmonic motion. Consider a free undamped
spring/mass system for which the spring constant is, say, k =10 lb/ft. Determine those masses mn that
can be attached to the spring so that when each mass is released at the equilibrium position at t = 0 with
a nonzero velocity v0, it will then pass through the equilibrium position at t =1 second. How many times
will each mass mn pass through the equilibrium position in the time interval 0<t<1?
Q. No. 8. Discuss the motion of a particle projected vertically upwards under gravity with initial velocity
U when the air resistance is proportional to the square of the velocity. With what velocity will the
particle return to earth?
Q. No. 9. Suppose the population of the world now is 4 billion and its doubling period is 35 years, what
will be the population of the world after 350 years, 700 years, 1050 years? If the surface area of the
earth is 1,860,000 billion square feet, how much space would each person get after 1050 years?
Q. No. 10. Substances X and Y combine in the ration 2:3 to form Z. When 45 grams of X and 60 grams of
Y are mixed together, 50 grams of Z are formed in 5 minutes. How many grams of Z will be formed in
210 minutes? How much time will it ake to get 70 grams of Z.
t
1?