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)

Show that the set of all solutions to the differential equation

f′′+f=0,

where f∈F(R), is a real vector space with respect to the usual definition of vector addition and scalar multiplication of functions

There is a patch of lily pads in a lake. Every week the patch doubles in size. It takes 45 weeks for the patch to cover the entire lake.

What if it takes 31 weeks for the patch to cover the entire lake, how many weeks would it take for the patch to cover 3.125% of the lake? (Round your answer to the nearest whole number)

If p(z) is a polynomial of degree n and that if α is a root of p(z) = 0, then p(z) factors as p(z) = (z−α)q(z) where q(z) has degree (n − 1). Use this and induction to show that a polynomial of degree n has at most n roots.

What kind of conic section (or pair of straight lines) is given by the quadratic form? Transform it to principal axes. Express xT = [x1 x2] in terms of the new coordinate vector ?T = [y1 y2] Show details.   

x1^2 + x1x2+x2^2=10

Maximizing Profit

The total daily revenue (in dollars) that a publishing company realizes in publishing and selling its English-language dictionaries is given by

R(x, y) = −0.005x² − 0.003y² − 0.002xy + 20x + 15y

where x denotes the number of deluxe copies and y denotes the number of standard copies published and sold daily. The total daily cost of publishing these dictionaries is given by

C(x, y) = 6x + 3y + 240

dollars. Determine how many deluxe copies and how many standard copies the company should publish each day to maximize its profits. (Round your answers to the nearest whole number of copies.)

deluxe copies =

standard copies =

What is the maximum profit realizable? (Round your answer to the nearest cent.)

$ _____

​Formulate the linear programming model for the issue​ algebraically by stating the:

1) Definition of Decision Variables,
2) Objective Function Equation,
3) Constraints Equations.

​​Set up and solve the problem using Excel.

Provide your answers and recommendations to the Manager of the company

General organization and formatting.

1. Richelieu Specialty Paints

Richelieu Specialty Paints is a company run by Amanda Richelieu. She is an artist and earned a BA degree with a major in Art and Design. The company was founded by Amanda’s father and he recently asked her to take over managing it so that he could retire. Amanda agreed and left her interior design job with an architecture firm to assume the role. She is keen to develop the business further.

The company makes a wide variety of paints, mostly targeted at fine artists. However, since Amanda has had experience in interior design she has determined that the company could apply its skills to develop new products for that industry and others.

She knows that she hasn’t yet developed her quantitative analysis skills to the point they need to be to run a company, so she is seeking assistance with four decisions that she needs to make. She would like to have a report from you providing recommendations for each decision, along with the background calculations that support your decisions.

Decision 1: Marketing - Advertising Placement

Amanda has a yearly advertising and promotions budget of $50,000. Her goal is to maximize the reach (the number of customers that see her company’s ads or other marketing efforts).

She is considering a mix of placing ads on art websites and in printed art magazines, as well as using targeted social media ad placements and appearances at art tradeshows. She wants to make sure that the company has an online presence, so would like to see at least 100 ads placed in any combination of websites and social media per year. However, she has data that suggests that if she advertises on the websites more than 60 times per year she will be wasting money. There are seven art tradeshows that the company could attend, but Amanda feels it is mandatory to attend at least the two most popular tradeshows. There are three major art magazines, each published monthly, that Amanda would like to advertise in. She could advertise in each of these once per month but would be comfortable reducing the frequency to a minimum of once every three months in each magazine and rotating the placements between the magazines.

The reach and cost of each type of ad placement or event is shown below:

• ● Recommend the appropriate mix of advertising and promotions for the company.

• ● Also comment on what would change if Amanda doubled her advertising and promotions

  budget. Estimate the increased reach if the budget were to double.

Decision 2: Financial - Assets Investment

Richelieu Specialty Paints has been quite successful and has $1,500,000 cash to be invested. Amanda and her management team have developed four potential investments in addition to leaving the cash in a low interest deposit. The four new opportunities are:

1. Purchase some shares of a major supplier (25% risk of total loss, 13% expected return)
2. Invest in the research and development of a new product (50% risk of total loss, 60% expected

return)
3. Buy a selection of corporate bonds (10% risk of total loss, 6% expected return) 4. Buy a selection of government bonds (1% risk of total loss, 3% expected return)

Leaving the money in a low interest deposit is virtually riskless and has a 0.5% return.

Richelieu’s father had provided Amanda with some guidelines for investment, aimed at diversification and risk management. These are:

1. Invest a maximum of 40% of the total cash in any one of the four new opportunities to ensure some risk control.

2. Invest at least $100,000 in each of the four new opportunities to ensure some diversity.

3. Limit the total potential loss to $300,000, based on the risk factors indicated.

4. Ensure the highest return possible.

• ● Recommend how much money should be invested in each of the four new investments, as well as the low interest deposit.

• ● Also, comment on how this mix would change, and what would happen to potential return, if Amanda decided to take a more conservative approach and limit the total potential loss to $150,000.

  Decision 3: Operations - Materials Blending

  Richelieu Specialty Paints has begun making paints for the interior design market. The company starts the process by blending four base paint mixes together that they buy from major manufacturers. They then add other components to finish off the paint mix to customer specifications. Each of the four base mixes includes pigment, solvent and three additives. When Richelieu blends the four base mixes together, they want the final product to be within these specifications (these numbers are a percentage of total volume):

• ● Pigment: 25% ≤ x ≤ 30%

• ● Solvent: 60% ≤ x ≤ 65%

• ● Additive1:3%≤x≤5%

• ● Additive2:5%≤x≤7%

• ● Additive3:1%≤x≤3%

The specifications of each of the four base mixes before they are blended together is:

• ● Recommend what percentage of each base mix should be included in the blend to achieve specifications listed, keeping input costs as low as possible.

• ● Base mix 2 is purchased overseas and thus the price fluctuates due to currency exchange rates. Comment on how the percentage of each base included in the blend would change if the cost for mix 2 rises to $35 per unit and note what would happen to the total cost in this situation.

  Note: you do not need units of volume, use the percentages. The final ‘blend’ needs to include some percentage of each of the four base mixes, adding up to a total of 100%.

  Decision 4: Operations and Marketing - Product Mix

  Richelieu Specialty Paints has received a special order for two types of paints: Marine Coat, used on boats and Blocker, used on items that receive a great deal of strong, hot sun. Both paints use a UV blocking additive and a mold inhibitor. Richelieu currently has 240 litres of the mold inhibitor on hand and 300 litres of the UV blocking additive.

  The special-order customer has requested at least 200 cans of Marine Coat but will take up to 450 cans if available. The customer will take as much Blocker paint as Richelieu can provide.

  Amanda would like to generate the highest total contribution margin possible for this order and needs to know how much of each product should be made.

Prove that the algorithm for computing the coefficients in the Newton form of the interpolating polynomial involves n^2 long operations (multiplication and division).