Category: Mathematics

discussion post

Respond to the following in a minimum of 175 words:

  • This week we began to examine relationships between quantities. Some ways we can compare quantities are by using percentages, ratios, rates, and unit labels. In your life, there are likely many situations that require you to compare quantities, understand the meaning of those comparisons, and make decisions based on those comparisons. What is an example of a situation from your professional or personal life that requires you to compare, understand, and make decisions based on quantitative comparison? Be sure to describe the types of quantitative comparisons you had to make, what decisions you made, and why. Would your decision be different if you had the benefit of learning this week’s concepts? Why or why not?

PHY 1001 Unit Five –Momentum

PHY 1001

Unit Five –Momentum

Mathematical Assignment

1. Calculate the momentum of a 265 kg motorcycle traveling at 25 m/s.

Key Idea  Stock of Data and

Solution

Sanity Check

2. A rubber ball (1 kg) falls from a height of 10 m and collides elastically with the floor.  What is the velocity of the ball as it rebounds from the floor?

Key Idea  Stock of Data and

Solution

Sanity Check

3. Two clay balls collide and stick. Ball 1 has a mass of 10 kg and an initial velocity of 10 m/s in the positive x direction. Ball 2 has a mass of 20 kg and an initial velocity of 5 m/s in the negative x direction.   What is the final velocity of the combined mass?

Key Idea  Stock of Data and

Solution

Sanity Check

4. Two clay balls collide and stick. Ball 1 has a mass of 10 kg and an initial velocity of 5 m/s in the positive x direction. Ball 2 has a mass of 20 kg and an initial velocity of 5 m/s in the negative x direction.   What is the final velocity of the combined mass?

Key Idea  Stock of Data and

Solution

Sanity Check

5. Two rubber balls undergo an elastic collision. Ball 1 has a mass of 10 kg and an initial velocity of 5 m/s in the positive x direction. Ball 2 has a mass of 10 kg and is stationary.   What is the final velocity of the two balls, v1f, and v2f?

Key Idea  Stock of Data and

Solution

Sanity Check

6. A ceiling fan turns at a rate of 20 RPM what is its speed in radians per second, and the speed of a tip of a 50 cm fan blade in m/s?

Key Idea  Stock of Data and

Solution

Sanity Check

7. Calculate the angular momentum of a 265 kg motorcycle traveling at 25 m/s. traveling around a circular curve 500 m in radius

Key Idea  Stock of Data and

Solution

Sanity Check

8. A ceiling fan turns at a rate of 20 RPM.  A 1 gram bug hangs on to the end of a 50 cm fan blade.  What is the magnitude of the frictional force between the bug’s feet and the blade?

Key Idea  Stock of Data and

Solution

Sanity Check

9. A wrench 30 cm long is attached to a nut and bolt.  You exert a 15 N force on the wrench.  What is the maximum torque you can exert under these conditions?

Key Idea  Stock of Data and

Solution

Sanity Check

10. A uniform beam of length 10 m and mass 100 kg is supported by pylons on either end.  A 200 kg box is centered at 5 m from the left end of the beam.  What is the downward force on the left most pylon?

Key Idea  Stock of Data and

Solution

Sanity Check

MFM1P0- Culminating Activity

MFM1P0- Culminating Activity

You are a city planner, trying to get the bid for a new city space. This task requires a series of steps and the application of knowledge from the course. We will be working on this activity throughout the entirety of the course, and at the end you will present your proposal to the Mayor (Ms. Pell). There will be one task related to each unit of study. Each task will be assigned at the end of the unit, instead of a unit test. At the end of the course you will incorporate feedback from the design phase and finalize your project. You may make changes from the design phase to the final product (just as many city planners do!) You will present your findings in any way you prefer (a video presentation, an interview with the mayor, a powerpoint, a poster, or any other approved method. Be creative!) You will need to complete a full report with total costs of your design in order for it to be approved by the city. Use the following questions to organize your ideas and show your work. Use proper mathematical communication to ensure your findings are clear and easy to understand. Unit 1: Geometry Unit 2: Algebra Unit 3: Proportions and Ratios Unit 4: Plane Geometry Unit 5: Linear Systems

Unit 1: Summative Task Your community space must consist of a pool. The pool can be any shape you desire, but keep in mind you will need to be able to calculate the area of the base as well as the volume of the shape. Your pool is required to have a deep end and a shallow end. The deep end and the shallow end must be composed of two different shapes (ex. Semicylinder and a rectangular based prism). To adhere to city guidelines the shallow end of the pool must be 0.5m deep. The deep end of the pool can be between 1.5m deep to 2.5m deep, you can decide what depth you would like it to be. 1. Draw your pool. Be sure to label all dimensions. [3A] 2. You must put a non-slip border around your pool. Calculate how many meters of non slip border you will be required to purchase. [4A] 3. The non-slip border costs $8.75/m. How much will it cost to purchase? [2T]

4. Your pool will need a cover. Determine how much area (hint: this area of the base) you will need to cover your pool. [5A] 5. If the material for a pool cover costs $5.00/m​2​, how much will it cost to cover your pool? [2T] 6. You will need to fill up the pool with water. How many liters of water will your pool hold? [6A] (hint: 1m​3​ = 1000L of water) [2T]

Self Assessment Checklist Use this little checklist to ensure you have meet the expectations!

Have I answer each question

Have I used formulas, and shown all of my work

Do my solutions have units

Have I reviewed material if questions were challenging

 

Have I included therefore statements for each questions

 

Have I double checked my calculations

maths

Please answer all questions asked in your initial post using detail and complete thoughts.

  • Scientific notation is used all around us, though it is generally hidden in metric prefixes. In the grocery store we might see it as a label saying there are 6060 milligrams of sodium in a product. This means 60×10−360×10−3 or 6×10−26×10−2 grams. What is an example of scientific notation you see in everyday use and what is one you might encounter in your major?

Problem Sheet 1 Mathematics for AI

Problem Sheet 1 Mathematics for AI

Problem Sheet 1 Week 2

This problem sheet consists of two questions. Each question contains three parts.

• Parts a), worth 40% of the marks in each question, test your knowledge of the core

material, and you should aim to provide good answers to this part of all questions.

• Parts b), worth 30% of the marks in each question, involve taking the concepts you have

been taught, but applying them in ways you may not have directly been shown. You

should attempt all the parts b), but you can still get a good mark without completing

all of them.

• Parts c), worth 30% of the marks in each question, are difficult questions which will test your understanding of the concepts taught in unfamiliar situations.

Question 1 (50%):

a) Translate the following sentences into propositional logic. Your formalizations should

be as detailed as possible.

i) Alice will go to the cinema or the theatre.

ii) Two is a prime number and not an odd number.

iii) If the speed limit is 30mph and I am driving at 25mph, then I am not breaking the

law.

iv) If Bob is not sleeping then he is working, eating, or relaxing.

v) Carlos will go to the park only if it is not raining.

Translate the following sentences into predicate logic. Your formalizations should be as

detailed as possible.

vi) Everyone is mortal.

vii) Unicorns do not exist.

viii) Every professional tennis player could beat any amateur tennis player.

ix) Everyone has either a father or a mother.

x) Somebody has visited every country that currently exists.

b) Consider the following formula of propositional logic:

P = ((A∧B)∨(¬A∧¬B)) →C

i) Suppose A = “I will go to the shops”, B = “I will go out for lunch” , C = “My

partner will be unhappy”. Translate P into an English sentence.

 

 

Problem Sheet 1 Mathematics for AI

ii) Suppose A is true, B is false and C is true. Is P true or false? Briefly explain

why.

iii) Which truth values for A , B and C result in P being false?

c) Consider the sentence “There is an animal in the zoo such that, if that animal is sleeping,

then every animal in the zoo is sleeping.”, formalised in predicate logic by the sentence

Q = ∃x(P x →∀yP y)

i) Suppose the zoo contains two animals, and both are sleeping. Is Q true or false?

ii) Now suppose one animal is sleeping, and one is awake. Is Q true or false?

iii) Now suppose we still know that the zoo contains two animals, but we do not know

how many are sleeping or awake. Can we know if Q is true or false?

iv) Now suppose we don’t know how many animals the zoo contains, except that there

is at least one animal, and we don’t know how many are awake or asleep. Can we

know if Q is true or false?

Question 2 (50%):

a) Identify which of these relations are reflexive, which are symmetric, which are transitive,

and which are equivalence relations. You do not need to show any working.

i) A is the set of all animals, R1 = {(a,b) |a is the same species as b}⊆A×A

ii) R2 = {(m,n) |m2 ≤n2}⊆Z×Z

iii) R3 = {(x,y) |x + y < 1}⊆R×R

iv) B = {0,1,2} , R4 = {(0,0),(0,1),(1,0),(1,1),(2,2)}⊆B×B

v) B = {0,1,2} , R5 = {(0,1),(1,2),(2,0)}⊆B×B

b) We define a function f : N→Z such that:

f (x) =

 x 2 if x is even

−x+12 if x is odd

i) Prove that f is an injection.

ii) Prove that f is a bijection.

iii) Given that f is a bijection, find the inverse function f −1 : Z→N

iv) Given the previous parts, what can we say about the cardinality of N and Z ?

c) In this problem sheet, we will call a relation R geometric if ∀x,y,z(xRy∧xRz →yRz)

i) Prove that if a relation is geometric and reflexive, then it is also symmetric.

 

 

Problem Sheet 1 Mathematics for AI

ii) Using the previous part, prove that if a relation is geometric and reflexive, then it

is an equivalence relation.

iii) Let f : A→B be a function. We define the relation

R⊆A×A, R = {(x,y) | f (x) = f (y)}

By showing that it is geometric and reflexive, show that R is an equivalence

relation. What are the equivalence classes of R ?

math questions

Work the following math problem

Question 1

Duke takes a car in for basic service. The service agent says a few extra repairs are needed, so Duke adds the cost of those repairs mentally, rounding to the nearest 10. What is Duke’s total estimate for the repairs? The costs are as follows:

  • Wheel alignment: $82
  • Transmission fluid flush: $157
  • Cabin air filter: $58

Note: 4 or less rounds down, 5 or more rounds up. For example, 14 becomes 10, while 15 becomes 20.

A. 280   B. 290   C. 300   D. 310

Question 2

Many gas stations give a discount for using cash instead of a credit card. A gas station gives a discount of 10 cents per gallon. William plans to pump 14 gallons. How much will William save by paying cash instead of credit card?   A. 10 cents   B. 24 cents   C. 100 cents   D. 140 cents

Question 3

A new company president is said to have caused the company “to do a 180.” Before the new president, the company was losing money. What is the company most likely doing under the new president?   A.  Losing a lot more money   B. Losing a little more money   C. Losing the same amount of money   D. Making money rather than losing

Question 4

Mo is on a baseball team and hears that a ball thrown at a 45 degree angle from the ground will travel the furthest distance. How should Mo release the ball for the furthest travel?   A. Nearly straight ahead, parallel to the ground   B. About halfway between straight ahead and straight up   C. About 2/3 of the way straight up   D. Nearly straight up, directly above his head

Question 5

One rule of thumb in the fast-food restaurant business is a “4 times markup”: The price of a food item should be four times the price of the ingredients used in making the item. If the cost of ingredients used in making a taco is 1.5 dollars, what should be the price of the taco?   A. 6 dollars   B. 7.5 dollars   C. 5.5 dollars   D. 4 dollars

Question 6

Alex invests $2,000 in a company’s stock. After a year, the value of Alex’s stock has increased to $2,500. What rate of return has Alex received?   A. 50%   B. 80%   C. 25%   D. 11%

Question 7

Travel Ez sells dollars at a rate of ($1.40)/(1 euro) and buys dollars at a rate of ($1.80)/(1 euro). At the beginning of a trip, Sophie exchanged $540 to get 300 euros. At the end of the trip she is left with 40 euros, so she exchanges the 40 euros back to dollars. How many dollars will Sophie get in exchange?   A. $72   B. $22   C. $56   D. $28

Question 8

Ryan remembers numbers using images that look somewhat like each number: 0 is a ball, 1 is a stick, 2 is a hanger, 3 is a comb, 4 is a kite, etc. Ryan remembered a 4-digit phone extension with this story: A person uses a hanger to pop a ball, then flies two kites. What number is Ryan likely remembering?   A. 2,044   B. 2,042   C. 2,004   D. 220

Question 9

Convert 2 3/4 to a decimal number.   A. 0.75   B. 1.50   C. 2.3   D. 2.75

Question 10

Consider a cookie recipe in which 1 1/2 cups of chocolate chips are needed to make 25 cookies. Lara wants to double the number of cookies. How many cups of chocolate chips should she use?   A. 1 1/2   B. 2   C. 2 1/2   D. 3

Question 11

Nik needs to estimate how many books will fit in a bin. Each book is 1 ft tall, 0.5 ft wide, and 0.1 ft thick. The bin is 5 feet wide, 2 feet tall, and 3 feet deep. Based on volume only, about how many books will fit in the bin?   A. 6   B. 66   C. 60   D. 60

Question 12

A person buys 28 bottles of orange juice for a party. The store clerk offers cardboard carrying cases for the bottles, each case holding 6 bottles. How many cases are needed, assuming each case should be filled as much as possible and the person will not be carrying any bottles outside of a case?  A. 1   B. 3   C. 4   D. 5

Question 13

A wall is 500 sq. feet. A gallon of paint covers 160 sq. feet. What is an appropriate conversion factor to help determine how many gallons will be needed to paint the wall?    A. (1 gallon)/(160 sq. feet)

B. (1 gallon)/(500 sq. feet)   C. (160 gallon)/(500 sq. feet)   D. (500 gallon)/(560 sq. feet)

Question 14

Jo is on the phone with a gardener who asks about the square feet of Jo’s backyard. Jo counts 20 steps long and 40 steps wide (with each step being about 3 feet). About how many square feet is the backyard?   A. 120 ft2   B. 360 ft2   C. 800 ft2   D. 7,200 ft2

Question 15

Zoey wants to use her iPad throughout a 6-hour flight. Upon takeoff, she uses the iPad for 2 hours and notices that the battery dropped by 25%, from 100% to 75%. How many total hours can Zoey expect from the iPad on a full battery charge? A.  10 hours   B. 4 hours   C. 8 hours   D. 6 hours

Question 16

The recipe for a fruit smoothie requires 2 cups of fruit per 6 cups of yogurt to maintain a 1/3 ratio. While making the fruit smoothie, Suzy accidentally puts 3 cups of fruit into the pitcher. How many cups of yogurt should be put in the pitcher to maintain the ratio of 1/3?   A. 3 cups   B. 9 cups   C. 1 cup   D. 5 cups

Question 17

Nik, a social worker for a county, helps county residents who are struggling with different issues. Nik logs the following hours meeting with clients (c) or doing other work (o):

  • Mon: 6 c, 4 o
  • Tue: 8 c, 2 o
  • Wed: 9 c, 1 o
  • Thu: 7 c, 3 o
  • Fri: Off

What percent of time did Nik spend with clients on Thursday?   A. 10%   B. 70%   C. 30%   D. 80%

Question 18

A person is holding two drill bits, one is 7/16 and the other is 25/64, and wants to first drill a hole using the smaller bit. To compare, the person expands 7/16 to 64ths. What is the expanded fraction?A. 14/64   B. 21/64   C. 24/64   D. 28/64

Question 19

A house is on an 80,000 sq. ft lot. About how many acres is the lot? There are 43,560 square feet in a acre.   A. 1/2 acre   B. 1 acre   C. 1 1/2 acres   D. 2 acres

Question 20

Sarah is planning a party at a party hall. The meal option is $50 per person and includes the hall for free. The hall-only option is $1,500 but allows an external caterer, which charges $30 per person. Sarah plans to invite 40 people. Which is a better cost comparison?   A. Meal option: $2,700. Hall-only option: $2,000. Hall-only option is cheaper.   B. Meal option: $1,000. Hall-only option: $3,500. Meal option is cheaper.   C. Meal option: $3,000. Hall-only option: $1,200. Hall-only option is cheaper.   D. Meal option: $2,000. Hall-only option: $2,700. Meal option is cheaper.

Question 21

A couple decides that Sophia will drive the first 3/5 of a trip and Lucas the last 2/5. The trip is 100 miles (the whole trip). How far will Sophia drive?   A. 40 miles   B. 60 miles   C. 20 miles   D. 30 miles

Question 22

The price of a sandwich decreases from $8 to $6. What is the percentage decrease in the price of the sandwich?   A. 33%   B.25%   C. 20%   D.14%

Question 23

A box is 24 inches long, 10 inches wide, and 10 inches deep. About how many cubic feet is the box?A. 1.4 ft3   B. 2.8 ft3   C. 240 ft3   D. 2,400 ft3

Question 24

A store in Minnesota advertises that on a holiday, everything is 20% off. A person buys shoes for $40 and socks for $10. In Minnesota there is no tax on shoes or socks. What is the final price?   A. $20  B. $40   C. $50   D. $30

Question 25

A self storage center is a storage room that is 8 feet long, 6 feet wide, and 10 feet high. What is the volume of the room?   A. 24 cubic feet   B. 48 cubic feet   C.140cubic feet   D. 480 cubic feet

Question 26

15 people are expected for a dinner. Three ounces of corn serves one person. A can of corn is nine ounces. How many cans of corn should be bought for the dinner?   A. 18 cans   B.12 cans   C. 3 cans   D. 5 cans

Question 27

6 people will attend a lunch. 2 cans of juice should be provided per person. Determine the total number of cans of juice required.   A.12 cans of juice   B. 6 cans of juice   C. 4 cans of juice   D.3 cans of juice

Question 28

Jules goes on a hike that will last 5 hours total. She brings 12 cups of water. After an hour, she has already drunk 3 cups of water. At that rate, how many cups would Jules need for all 5 hours?   A. 8 cups   B.15 cups   C. 9 cups   D.12 cups

Question 29

Dana is attaching a shelf to a wall and needs the shelf to be perpendicular to the wall. How many degrees should the shelf be relative to the wall?   A. 45   B. 90   C.180   D. 360

Question 30

A circular garden has a diameter of 12 feet. About how much trim is needed to surround the garden by placing trim on the garden’s circumference?   A. 38 feet   B. 48 feet   C. 144 feet   D. 432 feet

Math Discussion

his week focuses on an introduction to geometry. Examine your daily life, and look for examples of the following types of triangles:

  • Right
  • Acute
  • Obtuse
  • Scalene
  • Isosceles
  • Equilateral

For example, yield signs are examples of acute triangles, and also examples of equilateral triangles.

Respond to the following in a minimum of 175 words:

  • Share your daily examples.
  • How might this exercise and information be used to assist someone as they try to relate the importance of geometry to their daily life?
  • Provide an example of a problem you solve based on your chosen triangle example.

Technology In The Mathematics Classroom

As education advances in the 21stst century, educators must be prepared to use technology to engage students in learning, enhance instruction, and support students with exceptionalities.

Create a 7-10 slide digital presentation for a staff development session for elementary school teachers  on the use of technology in the mathematics classroom.

Your presentation should include:

  • Three technologies that can be used in the mathematics classroom to enhance instruction . Technology incorporated could include apps, computer programs, videos, websites, etc. Your choices should represent varied technologies. For each technology, include the following:
    • A detailed description of the technology including the features of the technology, how it works, and the elementary grade levels where it is developmentally appropriate
    • Benefits of using the technology in a mathematics classroom and how it fosters students’ engagement, communication, and confidence in using technology.
    • A mathematics classroom example of the use of the technology. Explain how teachers and students would interact with the technology for a specific math lesson
  • One assistive technology to support the communication and learning of students with communication disabilities.
  • A title slide, presenter’s notes, and a reference slide

The digital presentation should include graphics that are relevant to the content, visually appealing, and use space appropriately.

The Scenario: Park Furnishings Manufactures School And University Classroom Furniture

Park Furnishings manufactures school and university classroom furniture. The company has two production plants, located at Easton and Weston. The Easton plant produce tables and chairs and the Weston plant produces desks and computer workstations. Park has a working day of 7.5 hours and employs twenty workers at each plant. You may assume that there is a total of 24 working days every month.

All Park products are manufactured using finished pressed wood and polished aluminium fittings. Including scrap, each table and desk uses 20 m2 of pressed wood whereas each chair and workstation needs 12.5 m2 and 30 m2 respectively. A total of 240000 m2 of pressed wood is available each month and is divided equally between the two plants. The aluminium fittings that reinforce the legs of all the production items are purchased from an outside supplier. Each plant can purchase up to 9500 boxes of fittings per month and one box is required for every item of furniture produced. Production times are 72 minutes per table, 18 minutes per chair, 90 minutes per desk and 2 hours per workstation.

The unit profit for chairs and tables are £39 and £72 respectively, whereas each desk and workstation generates profits of £105 and £142 respectively.

Park is considering combining operations of both plants into a single factory. This consolidation will have the advantage of combining all available production resources as well as reducing administration costs by £1420 per month; however they have estimated that the cost of renovating and equipping the factory will be £1100000. Due to the current financial position Park Furnishings is only prepared to go ahead with the combined operation if it saves money in the first year of operation.

 

Let

· x1 represent the number of tables made per month;

· x2 represent the number of chairs made per month;

· x3 represent the number of desks made per month;

· x4 represent the number of workstations made per month;

where x1,x2,x3,x4 ≥0

 

(a) Easton

Enter the constraints for the Easton plant and the expression to be optimised.

Maximise:  x1 +  x2

subject to

x1 +  x2≤      [Wood]

x1 +  x2≤      [Metal fittings]

x1 +  x2≤     [Labour]

Sketch the constraints and hence find the values of

· (i) a,b,c, the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x1-axis;

· (ii) d,e,f,the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x2-axis;

Enter the values, to the nearest integer in the appropriate boxes below:

Enter a:
Enter b:
Enter c:
Enter d:
Enter e:
Enter f:

Now draw a sample profit line on your graph. Choose a value of the profit (P>0) and using this value, find the values of

(i) g, the intersection of your sample profit line with the x1x1-axis;

(ii) h, the intersection of your sample profit line with the x2x2-axis;

Enter the values, to the nearest integer in the appropriate boxes below:

Enter P:
Enter g:
Enter h:

Determine the optimal solution for x1 and x2 to the nearest integer and the profit that this solution will generate and enter your solution below.

The optimal solution is x1=  ,     x2=

Profit: £

Select the two constraints which intersect to give the optimal solution.

The optimal solution is the intersection of   Select Wood Metal Fittings Labour x1 ≥ 0 x2 ≥ 0  with   Select Wood Metal Fittings Labour x1 ≥ 0 x2 ≥ 0

 

(b) Weston

Enter the constraints for the Weston plant and the expression to be optimised.

Maximise:  x3 +  x4

subject to

x3 +  x4≤      [Wood]

x3 +  x4≤      [Metal fittings]

x3 +  x4≤      [Labour]

Sketch the constraints and hence find the values of

· (i) a,b,c, the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x3-axis;

· (ii) d,e,f, the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x4-axis;

Enter the values, to the nearest integer in the appropriate boxes below:

Enter a:
Enter b:
Enter c:
Enter d:
Enter e:
Enter f:

Now draw a sample profit line on your graph. Choose a value of the profit (P>0) and using this value, find the values of

· (i) g, the intersection of your sample profit line with the x3x3-axis;

· (ii) h, the intersection of your sample profit line with the x4x4-axis;

Enter the values, to the nearest integer in the appropriate boxes below:

Enter P:
Enter g:
Enter h:

Determine the optimal solution for x3 and x4 to the nearest integer and the profit that this solution will generate and enter your solution below.

The optimal solution is x3=  ,     x4=

Profit: £

Select the two constraints which intersect to give the optimal solution.

The optimal solution is the intersection of   Select Wood Metal Fittings Labour x3 ≥ 0 x4 ≥ 0  with   Select Wood Metal Fittings Labour x3 ≥ 0 x4 ≥ 0

 

(c) Combined

Enter the constraints for combining the plants and the expression to be optimised.

Maximise:  x1 +  x2 +  x3 +  x4

subject to

x1 +  x2+  x3 +  x4≤   [Wood]

x1 +  x2+  x3 +  x4≤  [Metal fittings]

x1 +  x2+  x3 +  x4≤  [Labour]

Determine the optimal solution for x1,x2,x3,and x4 and the profit that this solution will generate and enter your solution below. ( Enter the optimal solution correct to 3dp and the profit to the nearest pound.)

The optimal solution is x1=  ,     x2=  ,     x3=  ,     x4=

Profit: £

Is it economically sensible to combine the two plants?   Select Yes No I do not know

The scenario

Park Furnishings manufactures school and university classroom furniture. The company has two production plants, located at Easton and Weston. The Easton plant produce tables and chairs and the Weston plant produces desks and computer workstations. Park has a working day of 7.5 hours and employs twenty workers at each plant. You may assume that there is a total of 24 working days every month.

All Park products are manufactured using finished pressed wood and polished aluminium fittings. Including scrap, each table and desk uses 20 m2 of pressed wood whereas each chair and workstation needs 12.5 m2 and 30 m2 respectively. A total of 240000 m2 of pressed wood is available each month and is divided equally between the two plants. The aluminium fittings that reinforce the legs of all the production items are purchased from an outside supplier. Each plant can purchase up to 9500 boxes of fittings per month and one box is required for every item of furniture produced. Production times are 72 minutes per table, 18 minutes per chair, 90 minutes per desk and 2 hours per workstation.

The unit profit for chairs and tables are £39 and £72 respectively, whereas each desk and workstation generates profits of £105 and £142 respectively.

Park is considering combining operations of both plants into a single factory. This consolidation will have the advantage of combining all available production resources as well as reducing administration costs by £1420 per month; however they have estimated that the cost of renovating and equipping the factory will be £1100000. Due to the current financial position Park Furnishings is only prepared to go ahead with the combined operation if it saves money in the first year of operation.

 

Let

· x1 represent the number of tables made per month;

· x2 represent the number of chairs made per month;

· x3 represent the number of desks made per month;

· x4 represent the number of workstations made per month;

where x1,x2,x3,x4 ≥0

 

 

(a) Easton

Enter the constraints for the Easton plant and the expression to be optimised.

 

Maximise:  x1 +  x2

 

subject to

x1 +  x2≤      [Wood]

x1 +  x2≤      [Metal fittings]

x1 +  x2≤     [Labour]

Sketch the constraints and hence find the values of

· (i) a,b,c, the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x1-axis;

 

· (ii) d,e,f,the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x2-axis;

 

Enter the values, to the nearest integer in the appropriate boxes below:

Enter a:  Enter b:  Enter c:  Enter d:  Enter e:  Enter f:

 

Now draw a sample profit line on your graph. Choose a value of the profit (P>0) and using this value, find the values of

(i) g, the intersection of your sample profit line with the x1x1-axis;

(ii) h, the intersection of your sample profit line with the x2x2-axis;

 

 

Enter the values, to the nearest integer in the appropriate boxes below:

 

Enter P:  Enter g:  Enter h:

Determine the optimal solution for x1 and x2 to the nearest integer and the profit that this solution will generate and enter your solution below.

The optimal solution is x1=  ,     x2=

Profit: £

Select the two constraints which intersect to give the optimal solution.

The optimal solution is the intersection of        with

 

(b) Weston

Enter the constraints for the Weston plant and the expression to be optimised.

Maximise:  x3 +  x4

subject to

x3 +  x4≤      [Wood]

x3 +  x4≤      [Metal fittings]

x3 +  x4≤      [Labour]

 

Sketch the constraints and hence find the values of

· (i) a,b,c, the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x3-axis;

 

· (ii) d,e,f, the intersections of the Wood, Metal Fittings and Labour constraint respectively with the x4-axis;

 

 

 

 

Enter the values, to the nearest integer in the appropriate boxes below:

Enter a:  Enter b:  Enter c:  Enter d:  Enter e:  Enter f:

Now draw a sample profit line on your graph. Choose a value of the profit (P>0) and using this value, find the values of

· (i) g, the intersection of your sample profit line with the x3x3-axis;

· (ii) h, the intersection of your sample profit line with the x4x4-axis;

Enter the values, to the nearest integer in the appropriate boxes below:

Enter P:  Enter g:  Enter h:

Determine the optimal solution for x3 and x4 to the nearest integer and the profit that this solution will generate and enter your solution below.

The optimal solution is x3=  ,     x4=

 

Profit: £

 

Select the two constraints which intersect to give the optimal solution.

The optimal solution is the intersection of        with

 

(c) Combined

Enter the constraints for combining the plants and the expression to be optimised.

Maximise:  x1 +  x2 +  x3 +  x4

 

 

 

 

 

 

subject to

x1 +  x2+  x3 +  x4≤   [Wood]

x1 +  x2+  x3 +  x4≤  [Metal fittings]

x1 +  x2+  x3 +  x4≤  [Labour]

 

Determine the optimal solution for x1,x2,x3,and x4 and the profit that this solution will generate and enter your solution below. ( Enter the optimal solution correct to 3dp and the profit to the nearest pound.)

The optimal solution is x1=  ,     x2=  ,     x3=  ,     x4=

 

Profit: £

Is it economically sensible to combine the two plants?

Applied Mathmatics

Consider the frictionless rod, i.e. β=0. The equation of motion becomes

m (d^2 r)/(dt^2 )-mω^2 r=-mg sin⁡(ωt)

with g=9.81 m/s^2 and a constant angular speed ω.

The rod is initially horizontal, and the initial conditions for the bead are r(0)=r_0 and r^′ (0)=v_0.

A)Analytically solve this initial value problem for r(t) B)Consider the initial position to be zero, i.e. r_0=0. Find the initial velocity, v_0, that results in a solution, r(t), which displays simple harmonic motion, i.e. a solution that does not tend toward infinity. C)Explain why any initial velocity besides the one you found in part B) causes the bead to fly off the rod. D)Given r(t) displays simple harmonic motion, i.e. part B), find the minimum required length of the rod, L, as a function of the angular speed, ω. E)Suppose ω=2, graph the solutions, r(t), for the initial conditions given here: r_0=0 and initial velocities of v_0=2.40, 2.45, 2.50, and the initial velocity you found in part B). Use 0≤t≤5

Project 2: A bead sliding along a rod

Simple Harmonic Motion

A bead is constrained to slide along a rod of length . The rod is rotating in a vertical plane with a constant angular speed, , about a pivot in the middle of the rod. The pivot allows the bead to freely slide along the rod, i.e. the pivot does not impede the movement of the bead.

Let denote the distance of the bead away from the pivot where can be positive or negative.

 

 

 

 

 

 

Free Sliding Bead

Applying Newton’s second law provides a balance of forces due to gravity, friction, centripetal acceleration, and linear acceleration. The equation resulting from these forces is

where is the mass of the bead, is the coefficient of viscous damping, is the constant speed of angular rotation, is the acceleration due to gravity, and is the distance between the pivot and the bead.

The rod is initially horizontal, and the initial conditions for the bead are and .

Equation of Motion

Consider the frictionless rod, i.e. . The equation of motion becomes

with and a constant angular speed .

The rod is initially horizontal, and the initial conditions for the bead are and .

Analytically solve this initial value problem for

Consider the initial position to be zero, i.e. . Find the initial velocity, , that results in a solution, , which displays simple harmonic motion, i.e. a solution that does not tend toward infinity.

Explain why any initial velocity besides the one you found in part B) causes the bead to fly off the rod.

Given displays simple harmonic motion, i.e. part B), find the minimum required length of the rod, , as a function of the angular speed, .

Suppose , graph the solutions, , for the initial conditions given here: and initial velocities of , and the initial velocity you found in part B). Use

Problem 1

Consider the frictionless rod, i.e. . The equation of motion becomes

with and a constant angular speed .

The rod is initially horizontal, and the initial conditions for the bead are and .

You will need to write an Improved Euler Method system solver to find and

Numerically solve for when , , and . Solve in the time interval . Use step sizes and compare your results. Also, compare your best numerical answers with your analytic answers from Problem 1 part E).

Numerically solve for when , , and is selected to give simple harmonic motion, i.e. Problem 1 part B. Use small step sizes, e.g. etc. Solve for the longest time interval that provides reasonable values for . Compare your results to the analytic solution that gives simple harmonic motion. What does this demonstrate about numerical solutions?

Problem 2