Reference no: EM133674270

Computational methods for Data Analysis

Question 1: Consider the following problem. A paint company has four different types of paint colors (called PC-1 to PC-4) with the following compositions:

% in type, by volume, of color components
Type	Red(R)	Blue(B)	Y ellow(Y )
PC - 1	5	3	4
PC - 2	7	6	5
PC - 3	2	1	3
PC - 4	1	2	1

They need to blend these four paint colors into a mixture for which the composition by volume is: R = 22/5%, B = 33/10%, Y = 19/5%

How should they prepare this mixture? To answer this question, we need to determine the proportions of the four paint colors PC - 1, PC - 2, PC - 3, PC - 4 in the blend to be prepared.

To model the problem, we need to define the decision variables, denoted by xj = proportion of PC - j by weight in the mixture, where j = 1 . . . 4. It is also the fact that the sum of the proportions of various ingredients in a blend must always be equal to 1, that is ∑₁⁴ xj = 1.

i) Write the mathematical model of the problem, and if the model comes with a system of linear equations, then convert it into the matrix form Ax = B.

ii) Calculate det(A) = |A|. What is your observation about the type of solution the system has (none, one/unique, or infinitely many)?

iii) Perform row operations on the augmented matrix A|B and find the solution to the system (if it exists). (Leave your answer in exact form (with fractions; no decimals!)). In addition, check your obtain solution.

Question 2: Find and classify the critical points off (x, y) = x²y - x² - 1/3y³

Hints: To find the critical points of f (x, y), you need to consider f_x = f_y = 0. The function has continuous second-order derivatives near the critical points so you can check Hessian H = fxx  fxy of f at critical points.

                                                            fxy  fyy

Question 3: Consider a warehouse with limited storage space and four different types of products available for stocking. The warehouse manager wants to determine the optimal number of items to stock to maximize the total revenue generated from sales. The table below shows the unit price and storage space requirements for each product:

Product	Unit Price ($)	Storage Space (sq.ft)
1	6	15
2	5	14
3	7	16
4	3	10

Find the mathematical model of the problem to determine the number of units of each product that should be stocked in the warehouse to maximize the total revenue while ensuring that the total storage 500 (sq.ft) used as an available capacity.