1. An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below.

                                                   Medium         Cost Per Ad        # Reached          Exposure Quality

                                                        TV                   700                    8000                         35

                                                      Radio                 240                    3000                         50

                                                Newspaper             360                    5000                         25

If the number of TV ads cannot exceed the number of radio ads by more than 3, and if the advertising budget is $25,000. Develop the linear programming problem and solve the model that will maximize the number reached and achieve an exposure quality if at least 1000. (Hint: You need 3 decision variables to develop this LP problem.)

Vector operators

Let A ∈ Rn×n be symmetric and positive definite and let C ∈ Rn×m. Show:,

rank(C^TAC) = rank(C),

Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )

Exercise 2.16: Consider the folowing data

HistoData

36     39     36    35    36    20    19

46     40     42    34    41    36    42

40     38     33    37    22    33    28

38     38     34    37    17    25    38

1. Find the number of clases needed to construct an histogram.
2. Find the class length.
3. Define nonoverlapping classes for a frequency distribution.
4. Tally the number of values in each class and develop a frequency distribution.
5. Draw a histogram for these data.
6. Develop a percent frequency distribution.

IMPORTANT NOTE: SHOW ALL THE CALCULATIONS - (YOU WILL NOT GET ANY CREDIT IF YOU DO NOT SHOW ALL THE CALCULATIONS)

• Write at least 2 references in APA style.

y"-2y'+y=cos2t

1. general solution of corresponding homongenous equation

2. particular solution

3.solution of initial value problem with initial conditions y(0)=y'(0)=0

Write carefully the proof of the dual Pappus theorem.

Use Jupyter together with the latools.py module to do the computations of the system :
32x + 2y + 4z +0.5t=15
48x + 12y + z + 27t = 100
40x + y + 8z + t = 10
a solution with non-negative values for the food intakes and try to come up with the solution that
is the most appetizing for you.

The New England Maple Sugar Company of Los Angeles prepares 2 types of maple syrup from maple flavoring and water – called maple base – and sugar.  One gallon of ExtraMaple syrup requires 2 gallons of maple base and 4 pounds of sugar, and each gallon of regular maple syrup requires 5 gallons of base and 2 points of sugar.  This week the company has 10,000 gallons of maple base and 8800 pounds of sugar.  Company records indicate that at most 1800 gallons of ExtraMaple syrup can be sold in a week and syrup that is not sold cannot be counted in this week’s sales figures.  There is a net profit of $3 per gallon of regular maple syrup and $5 per gallon of ExtraMaple syrup.  Apply linear programming to determine how many gallons of each type should be produced to maximize net profit this week.

List the constraints and the objective function.

1. Find a triangulation for P^2.

How would you explain what a Riemann Sum is to a classmate?

Consider a Cauchy-Euler equation x^2y''- xy' +y =x^3 for x>0.

a) Rewrite the equation as constant- coefficeint equation by substituting x = e^t.

b) Solve it when x(1)=0, x'(1)=1.

Give a numerical example of non routine decision, Determine the relevant costs for this non routine decision and discuss the analysis (quantitative and qualitative) required to make the decision ( Numerical example)