State two real-world problems where you think that mathematical modelling is the only approach to find the solution of the problem. Give 4 essentials for each of the problems. Why do you think that there is no other scientific alternative for the treatment of these problems.
Problem 1: Epidemic Forecasting
Description: Predicting the spread of infectious diseases like COVID-19 to allocate medical resources effectively and implement preventive measures.
Essentials:
1. Population Dynamics Modeling: Mathematical models can simulate how individuals interact and how diseases transmit within a population. This involves differential equations representing susceptible, infected, and recovered (SIR) individuals.
2. Data Integration and Calibration: Incorporating real-time data on infection rates, demographics, healthcare capacity, and social behaviors into the models to accurately predict future outbreaks.
3. Policy Evaluation: Models can __________ ___ ___ ________ __________.
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If a planet was suddenly stopped in its orbit, supposed circular, show that it will fall into the sun in a time which is times the period of the planet’s revolution.
Compare the phase diagrams of the systems:
i)
ii)
by locating the equilibrium points and sketching the phase paths.
Suppose that the previous forecast was 2090 and the actual value of the variable of interest for the period was 1985 and the oldest value of interest was 1955. Using the moving average technique based upon the most recent four observations find new forecast for the next period.
A monopolist sets a price ‘p’ per unit and the quantity demanded ‘q’ is given by the following relation:
Let there be a fixed cost of Rs.9 and a marginal cost of Rs.1 per unit.
i) Write the profit function of monopolist.
ii) For maximum profit, find the number ‘x’ of units produced. Also find the maximum profit.
iii) A potential entrant enters into the business of the monopolist. He believes that the monopolist will go on making ‘x’ units. Write the profit function of the entrant.
iv) For maximum profit of entrant, find the number ‘z’ of units produced.
v) Find the maximum profit of the entrant. Explain whether he should enter into the business or not.
vi) Find the profit made by the monopolist after the entrant has entered into business.
Consider the cubic total cost function
Assume that the price of q is 15 per unit. Find the output which yields maximum profit.
Explain the method of delineating the efficient frontier of a feasible region.
A parachutist, whose weight (actually mass) is 64 kg, drops from a helicopter 5000 m. above the ground. She falls towards the earth under the influence of gravity. Assume that the gravitational force is constant. Assume that the force due to air resistance is proportional to the velocity of the parachutist. The proportionality constant is sec when the parachute is closed, and is sec
when it is open. If the parachute does not open until 1 minute after the parachutist leaves the helicopter, after how many seconds will she hit the ground?