Problem 8-25 (Algorithmic)

Georgia Cabinets manufactures kitchen cabinets that are sold to local dealers throughout the Southeast. Because of a large backlog of orders for oak and cherry cabinets, the company decided to contract with three smaller cabinetmakers to do the final finishing operation. For the three cabinetmakers, the number of hours required to complete all the oak cabinets, the number of hours required to complete all the cherry cabinets, the number of hours available for the final finishing operation, and the cost per hour to perform the work are shown here:

For example, Cabinetmaker 1 estimates that it will take 45 hours to complete all the oak cabinets and 63 hours to complete all the cherry cabinets. However, Cabinetmaker 1 only has 35 hours available for the final finishing operation. Thus, Cabinetmaker 1 can only complete 35/45 = 0.78, or 78%, of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1 can only complete 35/63 = 0.56, or 56%, of the cherry cabinets if it worked only on cherry cabinets.

1. Formulate a linear programming model that can be used to determine the proportion of the oak cabinets and the proportion of the cherry cabinets that should be given to each of the three cabinetmakers in order to minimize the total cost of completing both projects.
   

   Let	O1 = proportion of Oak cabinets assigned to cabinetmaker 1
   	O2 = proportion of Oak cabinets assigned to cabinetmaker 2
   	O3 = proportion of Oak cabinets assigned to cabinetmaker 3
   	C1 = proportion of Cherry cabinets assigned to cabinetmaker 1
   	C2 = proportion of Cherry cabinets assigned to cabinetmaker 2
   	C3 = proportion of Cherry cabinets assigned to cabinetmaker 3

   Min

   O1

   +

   O2

   +

   O3

   +

   C1

   +

   C2

   +

   C3

   s.t.

   O1

   C1

   ≤

   Hours avail. 1

   O2

   +

   C2

   ≤

   Hours avail. 2

   O3

   +

   C3

   ≤

   Hours avail. 3

   O1

   +

   O2

   +

   O3

   =

   Oak

   C1

   +

   C2

   +

   C3

   =

   Cherry

   O1, O2, O3, C1, C2, C3 ≥ 0

2. Solve the model formulated in part (a). What proportion of the oak cabinets and what proportion of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost of completing both projects? If required, round your answers for the proportions to three decimal places, and for the total cost to two decimal places.
   

   	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
   Oak	O1 =	O2 =	O3 =
   Cherry	C1 =	C2 =	C3 =

   
   Total Cost = $  
   
3. If Cabinetmaker 1 has additional hours available, would the optimal solution change?
   
   Yes
   
   Explain.
   
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.
   
   
   
4. If Cabinetmaker 2 has additional hours available, would the optimal solution change?
   
   Yes
   
   Explain.
   
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.
   
   
   
5. Suppose Cabinetmaker 2 reduced its cost to $39 per hour. What effect would this change have on the optimal solution? If required, round your answers for the proportions to three decimal places, and for the total cost to two decimal places.
   

   	Cabinetmaker 1	Cabinetmaker 2	Cabinetmaker 3
   Oak	O1 =	O2 =	O3 =
   Cherry	C1 =	C2 =	C3 =

   
   Total Cost = $  
   
   Explain.
   
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.

The table to the right gives size and distance data for the planets at a certain point in time. Calculate the scaled size and distance for each planet using a 1 to 10 billion scale model solar system.

Neptune 48,400 km 4539 million km

Complete the following table.

​(Type integers or decimals rounded to the nearest tenth as​ needed.)

generate the following matrices with given rank and verify with the rank command. Include the Matlab sessions in your report as indicated.

A) is 8x8 with rank 3.

B) is 6x9 with rank 4.

C) is 10x7 with rank 5 .

Find the dimensions of the following linear spaces.
(a) The space of all 3×4 matrices

(b) The space of all upper triangular 5×5 matrices

(c) The space of all diagonal 6×6 matrices

8. The cardinality of S is less than or equal to the cardinality of T, i.e. |S| ≤ |T| iff there is a one to one function from S to T. In this problem you’ll show that the ≤ relation is transitive i.e. |S| ≤ |T| and |T| ≤ |U| implies |S| ≤ |U|.

a. Show that the composition of two one-to-one functions is one-to-one. This will be a very simple direct proof using the definition of one-to-one (twice). Assume that f is one-to-one from S to T and g is one-to-one from T to U. Then show that f ○ g must be one-to-one from S to U.

b. For sets S, T, U prove that |S| ≤ |T| and |T| ≤ |U| implies |S| ≤ |U|. Hint: Apply the definitions of |S| ≤ |T| and |T| ≤ |U| then use part a to construct a one-to-one function from S to U.

c. Is it possible for ? ⊊ ? and |S| ≤ |T| to be true at the same time? That would mean T is proper subset of S but the cardinality of S is less than or equal to the cardinality of T. If it is possible, give an example. If it isn’t possible, prove that it isn’t possible

1) a) From the set {-8, -2/3, 5i, √(-9), √2, 0, 3+3i, -2.35, 7}

i) List the set of          Natural Numbers

ii) List the set of          Integers

iii) List of the set of    Rational Numbers

vi) List the set of         Real Numbers                                               

b) i) -30 ÷ -6 - (-12 + 8) – 4 x 3 =                           ii) 10(-2) - (-6)4

                          (-10) – 6(-3) =       (4, 4 marks)

2) a) 1²/₅ x 2 ¹/₇ ÷ 1⁴/₅ =                                                                          

2. After winning from a lottery were divided in the ratio 2 : 3 : 5,

If the largest amount received was $150,000.xx

How much was:-         i) The smallest amount =

                                   ii) The Total amount of prize money =         (3, 2 marks)

3) a) i) Calculate the simple interest earned if a deposit of $ 900,000 is

                        left for 12 years at an interest rate of 3% .                           

          ii) Calculate the interest rate for a 12 years investment of $600,000 to gained

                         $180,000 interest.                                                                 

2.    Solve

                             2(3x – 2 )= 6x - 2    + 1

                                      3             4                           (6 marks )                                                                         

c) Solve the following pairs of simultaneous equations

                      3x + y = 7   and     2x - 2y = 2        

Use direct substitution to verify that y(t) is a solution of the given differential equation in Exercise Group 1.1.9.15–20. Then use the initial conditions to determine the constants C or c1 and c2.

17. y′′+4y=0, y(0)=1, y′(0)=0, y(t)=c1cos2t+c2sin2t

18. y′′−5y′+4y=0,   y(0)=1 , y′(0)=0,   y(t)=c1et+c2e4t

19. y′′+4y′+13y=0, y(0)=1, y′(0)=0, y(t)=c1e−2tcos3t+c2e−3tsin3t

27. The growth of a population of rabbits with unlimited resources and space can be modeled by the exponential growth equation, dP/dt=kP.

Write a differential equation to model a population of rabbits with unlimited resources, where hunting is allowed at a constant rate α.

Write a differential equation to model a population of rabbits with unlimited resources, where hunting is allowed at a rate proportional to the population of rabbits.

Write a differential equation to model a population of rabbits with limited resources, where hunting is allowed at a constant rate α.

Write a differential equation to model a population of rabbits with limited resources, where hunting is allowed at a rate proportional to the population of rabbits.

30. Radiocarbon Dating.

Carbon 14 is a radioactive isotope of carbon, the most common isotope of carbon being carbon 12. Carbon 14 is created when cosmic ray bombardment changes nitrogen 14 to carbon 14 in the upper atmosphere. The resulting carbon 14 combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis. Animals acquire carbon 14 by eating plants. When an animal or plant dies, it ceases to take on carbon 14, and the amount of isotope in the organism begins to decay into the more common carbon 12. Carbon 14 has a very long half-life, about 5730 years. That is, given a sample of carbon 14, it will take 5730 years for half of the sample to decay to carbon 12. The long half-life is what makes carbon 14 dating very useful in dating objects from antiquity.

Consider a sample of material that contains A(t) atoms of carbon 14 at time t. During each unit of time a constant fraction of the radioactive atoms will spontaneously decay into another element or a different isotope of the same element. Thus, the sample behaves like a population with a constant death rate and a zero birth rate. Make use of the model of exponential growth to construct a differential equation that models radioactive decay for carbon 14.

Solve the equation that you proposed in (a) to find an explicit formula for A(t).

The Chauvet-Pont-d'Arc Cave in the Ardèche department of southern France contains some of the best preserved cave paintings in the world. Carbon samples from torch marks and from the paintings themselves, as well as from animal bones and charcoal found on the cave floor, have been used to estimate the age of the cave paintings. If a particular sample taken from the Cauvet Cave contains 2% of the expected cabon 14, what is the approximate age of the sample?

Give a direct proof for the 2nd Isormorphism Theorem of bi-modules over rings.

(Differential Equations) Consider the differential equation xy’-x⁴y³+y=0

1. Verify that the function y = (Cx²-x⁴)^-1/2 is a solution of the differential equation where C is an arbitrary constant.

2. Find the value of C such that y(-1) = 1. State the solution of the initial value problem.

3. State the interval of existence.

Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x, y, z, and w in terms of the parameters t and s.)

(x, y, z, w) = ( )

*Last person who solved this got it wrong

1. Ross White (see Problem 9) wants to reconsider his decision of buying the brackets and is considering making the brackets in-house. He has determined that setup costs would be $25 in machinist time and lost production time, and 50 brackets could be produced in a day once the machine has been set up. Ross estimates that the cost (including labor time and materials) of producing one bracket would be $14.80. The holding cost would be 10% of this cost.

(a) What is the daily demand rate?

(b) What is the optimal production quantity?

(c) How long will it take to produce the optimal quantity? How much inventory is sold during this time?

(d) If Ross uses the optimal production quantity, what would be the maximum inventory level? What would be the average inventory level? What is the annual holding cost?

(e) How many production runs would there be each year? What would be the annual setup cost?

(f) Given the optimal production run size, what is the total annual inventory cost?

(g) If the lead time is one-half day, what is the ROP?

One-Way ANOVA and Multiple Comparisons

The purpose of one-way analysis of variance is to determine if any experimental treatment, or population, means, are significantly different. Multiple comparisons are used to determine which of the treatment, or population, means are significantly different. We will study a statistical method for comparing more than two treatment, or population, means and investigate several multiple comparison methods to identify treatment differences.

-Search for a video, news item, or article (include the link in your discussion post) that gives you a better understanding of one-way analysis of variance and/or multiple comparison methods, or is an application in your field of study.

-Explain in your post why you chose this item and how your linked item corresponds to our One-Way ANOVA and Multiple Comparisons course objectives.

-Then describe how you could use any of these methods in your future career or a life situation.

Three frogs are placed on three vertices of a square. Every minute, one frog leaps over another frog, in such a way that the "leapee" is at the midpoint of the line segment whose endpoints are the starting and ending positions of the "leaper". Will a frog ever occupy the vertex of the square which was originally unoccupied?

The n- dimensional space is colored with n colors such that every point in the space is assigned a color. Show that there exist two points of the same color exactly a mile away from each other.

Find the vector and parametric equations for the plane. The plane that contains the lines r1(t) = <6, 8, 8,> + t<-2, 9, 6> and r2 = <6, 8, 8> + t<5, 1, 7>.