- Mathematical Modelling,

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Are you looking to download a PDF soft copy of the Solved Assignment **MTE-14 - Mathematical Modelling**? Then GullyBaba is the right place for you. We have the Assignment available in English language.

This particular Assignment references the syllabus chosen for the subject of **Mathematics**, for the ** January 2020 - December 2020** session. The code for the assignment is MTE-14 and it is often used by students who are enrolled in the B.Sc. Degree.

Once students have paid for the Assignment, they can Instantly Download to their PC, Laptop or Mobile Devices in soft copy as a PDF format. After studying the contents of this Assignment, students will have a better grasp of the subject and will be able to prepare for their upcoming tests.

- a) Suppose that a given population can be divided into two parts: those who have a given disease and can infect others, and those who do not have it but are susceptibles. Assume that the disease spreads by contact between sick and well members of the population and the rate of spread is proportional to the number of such contacts. If the initial population of infectious individuals is 100 then

i) Formulate a mathematical model for the given problem and write a differential equation governing it.

ii) Find the equilibrium points of the differential equation.

iii) Solve the given problem. What happens to the spread of the disease as (time) t → ∞?

b) A particle of mass m is thrown vertically upward with velocity v0. The air resistance is mg cv2 where c is a constant and v is the velocity at any time t. Show that the time taken by the particle to reach the highest point is given by

v0 √c = tan (gt √c)

c) The respiratory flow of air in the lungs is affected due to air pollution. If you have to model respiratory flow write four essentials for the model. - a) A shell when projected at an angle of tan−1 1/3 to the horizon falls 60 m. short of the target. When it is fixed at an angle of 45˚ to the horizon, it falls 80m beyond the target. How far is the target from the point of projection?

b) Assume that a spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is 3 mm, and one-half hour later has been reduced to 2 mm, find an expression for the radius of the raindrop at any time.

c) Suppose a viscous oil, whose flow is in the laminar regime is to be pumped through a 10 cm diameter horizontal pipe over a distance of 15km at a rate of 10−3 m3 /s. Viscosity of the oil is 0.03 poise. What is the required pressure drop to maintain such a flow? - a) If a simple pendulum of length oscillates through an angle α on either side of the mean position then find the angular velocity dθ/dt of the pendulum where θ is the angle which the string makes with the vertical.

b) The population x(t) of a certain city satisfies the logistic law

where t is measured in years. Given that the population of the city is 100000 in 1980, determine the population at any time t >1980. Also find the population in the year 2000.

c) Find the output yielding maximum profit for the cost function

given that the cost price of x is Rs. 45/- per unit. - a) Given one example each from the real world for the following, along with justification, for your example

i) A non-linear model

ii) A linear, deterministic model.

b) Consider arterial blood viscosity µ = 0.025 poise. If the length of the artery is 5.1 cm, radius 8×10−3cm. and P = P1 – P2 = 4×103 dynes/cm2 then find (i) maximum peak velocity of blood and (ii) the shear stress at the wall.

c) Suppose a planet describes an ellipse with sun S as its focus, whose major axis is 2a and minor axis is 2b. Let P(x, y) be the position of the planet after time t after starting from rest from perihelion position A, referred to S as origin. Let θ be the eccentric angle of the position P.

Show that h t = ab (θ − esin θ) - a) Formulate a one-dimensional model describing the dynamics of phytoplankton growth C(x, t) in a water mass taking into account the following: D, its diffusion coefficient, α its rate of growth, β its mortality rate due to sinking. Fixing the area of interest as 0 ≤ x ≤ 1 and the initial concentration of phytoplankton as 20 moles cm/3, find the concentration distribution of phytoplankton in 0 ≤ x ≤ 1 at any time t.

b) Suppose you are driving a van down a highway. Use dimensional analysis to find the wind force you are experiencing, assuming that the force is affected by the wind density, the speed of the van and its surface area exposed to the wind direction. - a) Consider the following system of differential equations representing a prey and predator population model

i) Identify all the real critical points of the system

ii) Obtain the type and stability of these critical points.

b) A cassette player repairman finds that the time spent on his job has an exponential distribution with mean 15 min. If he repairs sets in the order in which they came in, and if the arrival of sets is approximately Poisson with an average rate of 18 per 9 hours a day, what is repairman’s expected idle time each day? How many jobs are ahead of the set just brought in?

c) Bacterial cells grow at a rate proportional to the volume of dividing cells at that moment. If V0 is the volume of dividing cells at any time t. Find the time at which the volume of the cells will be double its original size. - a) Two players A and B are involved in a game. Each player has three different strategies. The pay-off table is given below:

Find the saddle point and value of the game.

b) Newton’s law of cooling assumes that air at room temperature is blown past the cooling body (forced cooling). For cooling in still air (natural cooling) a better modal is to assume that the rate of temperature decrease of the cooling body is directly proportional to the 5/4th power of the difference between the temperature u of the body and the temperature us of the surrounding air.

i) Write the law for natural cooling as a differential equation. Is this equation linear?

ii) Solve the equation obtained in i) above assuming that initially, the temperature of the cooling body was u0.

c) Consider the following first-order ODE formulations

Associate the physical meaning to the variables {t, n(t)} and the parameters {a, L} so that the above formulation becomes a mathematical model for population changes. - a) The rate of increase of susceptible AIDS victims is proportional to the number of susceptible persons and number of infected persons. If there are S0 susceptible persons and 1 infected person at a time to then i) set up the equation for the spread of the disease ii) solve the resulting equation iii) give a physical interpretation to the same by plotting the epidemic curve iv) write the limitations of the model.

b) The sales of a company from 1993-1998 are given below:

Fit a linear curve using the least squares method. Hence find out the company’s sales in 1999. - a) Let ρ = (w1, w2) be a portfolio of two securities. Find the value of w1 and w2 in the following situations:

i) ρ12 = −1 and ρ is risk-free.

ii) σ1 = σ2 and variance P is minimum.

iii) Variance on P is minimum and ρ12 = −0.5, σ1 = 2 and σ2 = 3.

b) The cost of production of a substance per unit is given by the formula C = q2 − 4q + 1, where q is the material cost. Find the selling price per unit, so that the profit on 100 units will be Rs. 200, if q = 15. Also calculate the cost of material per unit so that profit of 100 units can be maximized, if the selling price is Rs. 200. - a) Two firms X and Y produce the same commodity. Due to production constraints, each firm is able to produce 1, 3 and 5 units. The cost of producing qx units for firm X is and firm Y has identical cost function for producing qy units. p is the price of one unit for firm X . We assume that the market is in equilibrium. The outcomes are the profits of the firm shown in the form of a matrix A = {aij}. Write (i) a11 (ii) a22 (iii) a21, if demand function D(p) is given as D( p) = 50 − p.

b) Assume that the moon is at a distance of 3,00,000 kms from the earth and that it takes 28 days for it to orbit the earth once. Geostationary satellites are those which are at a rest relative to earth. Using these two statements derive the altitude of the geostationary satellite from the centre of the earth.

- University IGNOU (Indira Gandhi National Open University)
- Title Mathematical Modelling
- Language(s) English
- Session January 2020 - December 2020
- Code MTE-14
- Subject Mathematics
- Degree(s) B.Sc.
- Course Core Courses (CC)
- Author Gullybaba.com Panel
- Publisher Gullybaba Publishing House Pvt. Ltd.

The IGNOU open learning format requires students to submit study Assignments. Here is the final end date of the submission of this particular assignment according to the university calendar.

**15th July 2021**(if Enrolled in the**July 2020**Session)**30th Sept. 2021**(if Enrolled in the**January 2021**Session).

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- January 2020 - December 2020 41 Pages (0.00 ), PDF Format SKU: IGNGB-AS-BS-MTE14-EN-

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