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MATH3290 Mathematical Modeling (2014-15)Assignment 3Due: Nov 14

1. Use the Monte Carlo simulation to find the volume trapped between the two paraboloids

z= 8 x2 y2 and z= x2 + 3y2

Note that the two paraboloids intersect on the elliptic cylinder

x2 + 2y2 = 4

Give also a 95% confidence interval.

2. A farmer owns 45 acres of land. She is planning to plant each acre with wheat orcorn. Each acre of wheat yields $200 in profits, whereas each acre of corn yields

$300 in profits. The labor and fertilizer requirement for each are provided in thefollowing table. The farmer has 100 workers and 120 tons of fertilizer. Determinehow many acres of wheat and corn need to be planted to maximize profits.

Wheat CornLabor(workers) 3 2Fertilizer(tons) 2 4

Table 1: Data set for problem 5.

Solve this problem by the Simplex method. In each step, clearly state the entering

and leaving variables, independent and dependent variables, and the current valueof objective function.

3. Solve the following by the Branch-and-Bound algorithm:Maximize x+ ysubject to

2x+ 5y 16

6x+ 5y 27

x 0, y 0, x , y are integers

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4. Your company sells a product whose demands over the next 4 months are 100, 140, 210and 180 units respectively. You can stock just enough supply to meet the demandeach month, or you can overstock to meet the demand for two or more consecu-tive months. In the latter case, a holding cost of $1.2 is charged per overstockedunit per month. You estimate the unit purchase prices for the next 4 months are$15, $12, $10 and $14 respectively. A set up cost of $200 is incurred each time a

purchase order is placed. Your company wants to develop a purchasing plan thatwill minimize the total costs of ordering, purchasing and holding an item in stock.

(a) Formulate this problem as a shortest path problem.

(b) Find the optimal solution by the Dijkstras algorithm.

5. The academic council is seeking representation from among 6 students who areaffiliated with 4 societies. The academic council representation includes three areas:mathematics, art and engineering. At most two students in each area can be onthe council. The following table shows the membership of the 6 students in the 4societies:

Society Students1 1,2,32 1,3,53 3,4,54 1,2,4,6

The students who are skilled in the areas of mathematics, art and engineering are

shown in the following table:

Area StudentsMathematics 1,2,4

Art 3,4Engineering 4,5,6

A student who are skilled in more than one area must be assigned exclusively to onearea only. Can all 4 societies be represented on the council? Show your calculations.

Reference: A First Course in Mathematical Modeling, by Giordano, Fox, Horton andWeir. 4th Edition