A profit maximizing monopolist produces a quantity of a commodity. His
D(p) = 30 , where p is the price. Find the values of p, q and the profit.
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The following table gives the sales of an item produced by a company since 1999.
| Year | Sales (in crores of rupees) |
| 1999 2000 2001 2002 2003 2004 | 10 12 9 13 12 14 |
Consider the population x(t) of fish and y(t) of sharks at any time t in a certain region of the pacific ocean. Sharks prey on fishes and their population would decrease if isolated from the fish population. Let us assume further the following assumptions.
1) The change in the shark and fish population, in isolation, is respectively proportional to the present population of sharks and fish.
2) The number of sharks and fish caught by fisherman is directly proportional to the present population of the shark and fish population respectively. The fishing methods do not discriminate between sharks or fish.
3) The number of fish eaten by sharks is directly proportional to the product of the number of fish present and the number of sharks present.
4) The additional number of sharks surviving is directly proportional to the number of fish eaten. Under assumptions 1) to 4)
i) Formulate the mathematical model for the given problem and write a
system of differential equations governing it.
ii) Is the system of equations obtained in i) an autonomous system? Give reasons.
iii) Find the steady-state solution of the system in i) above. What does the solutions obtained correspond to?
iv) From the system of equations obtained in i) obtain the expression for
dy / dx and hence solve it under the condition x(0) = x0 , y(0) = y0 .
In a perfectly competitive market, the supply function S(p) and demand function D(p) are given as follows:
, where p is the price:
i) Find the equilibrium prices
ii) Using Walrasian stability condition, determine whether the equilibrium prices are stable.
A particle moves from an initial velocity u at a constant acceleration f. Find an expression for the distance covered in time t using dimensional analysis.
Let a, K and 0 N be positive real numbers such that 0 < N0 <K. Find the solution of the logistic equation
If N0, N1, and N2 be the values of N at t = 0, t = T and t = 2T respectively, then show that
Also if at time t = t , the population N reaches half its carrying capacity then what is the form of the solution of the logistic equation?
Find the current density when oxygen diffuses through a red cell of thickness 2.5 10-8 cm , the two ends of which are maintain at a fixed concentration C0 .
Suppose we find the sum of solutions obtained in (i) and (ii) of (b) above. Does this give a solution of the equation describing the motion? If yes, what physical situation, this solution will be correspond to? When is the velocity of the pendulum maximum? What type of energy does the pendulum have at position of maximum velocity?