NIIIII MATH
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BINF 5020 Biomedical Modeling and decision making
1. Get the first derivative of the following functions sin x, cos x, and tan x.
2. Given :
y f(x)
x f(x)
obtain ,4 x )( 22
∂ ∂
∂ ∂
++=
and
yxyxf
( these are partial derivatives of function w.r.t. x and y.)
3. Solve the following differential equation:
dy/y = 4 dt , with y(0) = 2 at t = 0. Obtain the solution y(t)
4. Solve the following linear set of equations: x + 2 y = 5 2 x + 2 y = 8 obtain the solution for x and y.
5. Given
−−−
=
=
111 321
Y and 31 12 21
X
Obtain X T . Multiply these matrices to obtain X*Y. Can you multiply Y*X?
Obtain results. 6. Given the following numbers:
[ 2, 7, 9, 6, 8, 11, 2]
get the mean, median, variance and standard deviation.
7. A girl has a mass of 110 pounds (1 pound = 0.4536 kg). She eats a 2 ounce chocolate bar. The energy content in the chocolate is 4700 K cal/kg. Assume that all the energy from the chocolate can be converted into mechanical energy, answer the following: • Can she climb a hill of 2000 ft high? (1 cal = 4.1868 Joules). • Calculate the total distance she can she run at the speed of 6 miles per
hour on the energy provided by the 2 ounce of chocolate bar. • After eating the chocolate bar and climbing a hill of 2000 feet, the
remaining chocolate energy is totally converted into the fat at a rate of 9.0 K calories per gram of fat. Calculate the amount of fat gained by the girl.
8. Given a very simple population growth model where population increase is
proportional to current population. The model is given by:
dy(t)/dt = 2y, and the population at t = 0 is given as 10. Find the population at t = 20. Assume time units years.
9. An output of a model is specified by the following equation: y(t) = at e -bt ,with a = 4 and b = 2 Find its maximum value and at what time this maximum occurs. Show all the intermediate calculations.
10. Ten patients need heart transplant and four hearts for the transplant surgery are available. How many ways are there to make a list of recipients. How many ways are there to make a list of the six out of ten who must wait for further donors.
11. Suppose an ice-cream seller at a summer fair guesses the amount of ice-creams that he is going to sell. It is proportional to number of people in the fair, proportional to temperature in excess of 15 0 C, and inversely proportional to the square of the selling price. Develop an appropriate model for the number of icecreams to be sold in the summer fair.
12. Given a population growth model of fish population. Find the approximate
solution of using numerical technique:
dy/dt = 2.y 2 (t) - 3 y(t), where y(0) = 2. Calculate the population y(k) at time points k = 3,4,5.
13. The pressure p at the depth h below the surface of a fluid of density ρ is given by :
p = ρ g h, where g is acceleration due to gravity. Verify the dimensionality of the equation by performing the dimensional analysis.
( 4 marks) 14. A girl has a mass of 110 pounds (1 pound = 0.4536 kg). She eats a 2 ounce
chocolate bar. The energy content in the chocolate is 4700 K cal/kg. Assume that all the energy from the chocolate can be converted into mechanical energy, answer the following: • Can she climb a hill of 2000 ft high? (1 cal = 4.1868 Joules). • Calculate the total distance she can she run at the speed of 6 miles per
hour on the energy provided by the 2 ounce of chocolate bar. • After eating the chocolate bar and climbing a hill of 2000 feet, the
remaining chocolate energy is totally converted into the fat at a rate of 9.0 K calories per gram of fat. Calculate the amount of fat gained by the girl.
( 4 marks)
15. Given the modeling relationship: y(t) = 8 t - t2
(2 marks) Find out at what time points the value of the function will be zero and calculate the maximum value of the function y(t).
16. A large heard of wild animals lives in a region where the rainfall varies in a
periodic manner with maximum occurring every four years. The animals live by grazing and the amount of food depends on the rainfall. The net birth rate can be assumed to be proportional to the amount of food available, the maximum are being 2 % per annum. There is also a net emigration of 100 animals per annum form the region. The present size of the population is 5000 animals and prediction for the next 20 years are required. Develop a model for this problem
17. The rate of cooling of a hot body in the air is very nearly proportional to the difference between the temperature of the body and the temperature of the air. If the temperature of the air is 18 degree C and the body is initially at 60 degree C is found to have cooled to 50 degree C after 3 minutes, how long will it be before it cools to 30 degree C? What will be its temperature after 10 minutes?
18. A conical tank of height 2 m is full of water and the radius of the surface is 1 m. After 8 hours, the depth of the water is only 1.5 m. If we assume that the water evaporates at a rate proportional to the surface area exposed to the air, obtain the mathematical model for predicting the volume of the water in the tank after the time t. A variable w is related to two other variables x and y in such a way that w is proportional to x and also proportional to Y. Which of the following correctly expresses the relationship?
W = a(x+y), w =ax+by and w = axy ans ( c ) 19. When a fluid flows through a pipe, the frictional force F between the pipe wall
and the fluid is assumed to be proportional to the length L of the pipe and the square of the fluid velocity V. It is also assumed to be inversely proportional to the diameter D of the pipe. Write down an expression for the force F in terms of L, V, and D and involving constant k. What are the dimensions of k? In what units the k would be measured?
F = klv2/d [k] = ML-1 in kg/m
20. The army X is about to attack the army Y which has only 5000 troops while the army X has 10,000. The army Y, however, has superior military equipment which makes Y soldier 1.5 times as effective as an X solder. You wish to develop a mathematical model for the resulting battle and use the model for the following purpose. Assume that 0.15X soldiers are killed by each Y soldier in 1 hr and that 0.1 Y soldiers are killed by each Y soldier in I hour.
• To predict which army will win • To estimate how many troops of the winning army will be left at the
end • Calculate how many troops the loosing army would have needed
initially to win the battle.
21. When the earth is modeled as a sphere of diameter 12.72 x 10 x 3 km, it is obvious that the walls of a tall building are vertical, they can not be parallel. Suppose that a tower block is 400 m tall and the ground floor has an area of 2500 m 2 . How much extra area is there on the top floor? 0.314 m 2 .Make any necessary assumptions for calculations.
22. One estimate of the probability of a mutation at each nucleotide position in a single reproductive cycle is 10
-8 . In an organism with 10
7 nucleotides, what is the
probability that no mutation takes place?Assume that the probability of mutation of each nucleotide is same and they are independent.
What is the probability of at least one mutation.
23. An experiment was performed with pea plant in which two parents were crosses to get an F1 generation. ParentP1 had round yellow seed and plant P2 had wrinkled green seeds. The F1 plants all had round yellow seeds.. Let R = allele for round seeds, r = allele for wrinkled seeds, Y = allele for yellow color and y = allele for green color. Assume that genes for shape and color are independent. What can you say about dominance of R vs r and Y vs y in the F2 generation.
What are the probabilities of P(round,yellow), P(round ,green), P(wrinkled, yello), and P(wrinkled, green).
24. Suppose that we have a collection of N amino acid molecules with n 1 of type 1,
n 2 of type 2, …….n
20 of type 20, with sum of all n
i ’s equal to N. Obtain an
expression for the number of different types of proteins that can be obtained with those N molecules ( use all N for each type of protein).
25. Suppose that you have a chamber with 90 flies and you drop another 10 additional ones. You wait for 5 minutes and select a random sample of 10 flies. What is the probability that you will get the same 10 flies back?
26. Given four different types of molecules. How many different sequence can be formed of length 2, length 3, and length 4?
