Using Mathematics To Solve Real World Problems
Using Mathematics to Solve Real World Problems
Creating a mathematical model:
Creating a mathematical model: We are given a word problem
Creating a mathematical model: We are given a word problem Determine what question we are to answer
Creating a mathematical model: We are given a word problem Determine what question we are to answer Assign variables to quantities in the problem so that you can answerthe question using these variables
Creating a mathematical model: We are given a word problem Determine what question we are to answer Assign variables to quantities in the problem so that you can answerthe question using these variables Derive mathematical equations containing these variables
Creating a mathematical model: We are given a word problem Determine what question we are to answer Assign variables to quantities in the problem so that you can answerthe question using these variables Derive mathematical equations containing these variables Use these equations to find the values of these variables
Creating a mathematical model: We are given a word problem Determine what question we are to answer Assign variables to quantities in the problem so that you can answerthe question using these variables Derive mathematical equations containing these variables Use these equations to find the values of these variables State the answer to the problem
Creating a mathematical model: We are given a word problem Determine what question we are to answer Assign variables to quantities in the problem so that you can answerthe question using these variables Derive mathematical equations containing these variables Use these equations to find the values of these variables State the answer to the problemToday we will do this using straight lines as our equations, and wewill solve the problem by drawing these lines (graphing).
Creating a mathematical model: We are given a word problem Determine what question we are to answer Assign variables to quantities in the problem so that you can answerthe question using these variables Derive mathematical equations containing these variables Use these equations to find the values of these variables State the answer to the problemToday we will do this using straight lines as our equations, and wewill solve the problem by drawing these lines (graphing).This process is called “Linear Programming” and is one of the mostpowerful mathematical methods used by businesses and companies tosolve problems and help them make the best decisions.
Creating a mathematical model: We are given a word problem Determine what question we are to answer Assign variables to quantities in the problem so that you can answerthe question using these variables Derive mathematical equations containing these variables Use these equations to find the values of these variables State the answer to the problemToday we will do this using straight lines as our equations, and wewill solve the problem by drawing these lines (graphing).This process is called “Linear Programming” and is one of the mostpowerful mathematical methods used by businesses and companies tosolve problems and help them make the best decisions.“Operations Research” is the profession that applies mathematicalmethods like this to problems arising in industry, healthcare, finance,etc.
A problem:
A problem:A furniture manufacturer produces two sizes of boxes (large, small) that areused to make either a table or a chair.
A problem:A furniture manufacturer produces two sizes of boxes (large, small) that areused to make either a table or a chair.tablelarge blockchairsmall block
A problem:A furniture manufacturer produces two sizes of boxes (large, small) that areused to make either a table or a chair.A table makes 3 profit and a chair makes 5 profit.table 3large blockchairsmall block 5
A problem:A furniture manufacturer produces two sizes of boxes (large, small) that areused to make either a table or a chair.A table makes 3 profit and a chair makes 5 profit.If M small blocks and N large blocks are produced, how many tables andchairs should the manufacturer make in order to obtain the greatest profit?tableNlarge blockchairM 3small block 5
Large blockSmall blockTable1 large block1 small blockChair1 large block2 small blocks
Problem: Given M small blocks and N large blocks, howmany tables and chairs should we make to obtain themost profit?Profit: 3Table 5Chair
Example: 12 small blocksand 12 large blocks1212
12Example: 12 small blocks12and 12 large blocksWe can make 4 tables and 4 chairs:44
12Example: 12 small blocks12and 12 large blocksWe can make 4 tables and 4 chairs:44 large4 small44 large8 small8 large12 small
12Example: 12 small blocks12and 12 large blocksWe can make 4 tables and 4 chairs:44 large4 small44 large8 smallProfit ( 3) x 4 ( 5) x 4 328 large12 small
1212 small blocks1212 large blocks4 tables and 4 chairs: 34 54Used:8 large blocks12 small blocks(4 large blocks left)
1212 small blocks1212 large blocks4 tables and 4 chairs: 34 54Used:8 large blocks12 small blocks(4 large blocks left)I can make 2 more tables if I make 1 less chair; 3 chairs and 6 tablesÆ increase my profit! (1 chair Æ 2 tables, profit goes up by 1)Profit ( 3) x 6 ( 5) x 3 33
1212 small blocks1212 large blocks6 tables and 3 chairs: 36 53Used:9 large blocks12 small blocks(3 large blocks left)
1212 small blocks1212 large blocks6 tables and 3 chairs: 36 53Used:9 large blocks12 small blocks(3 large blocks left)I can do it again; change one chair into 2 tables; 8 tables and 2 chairsProfit ( 3) x 8 ( 5) x 2 34
1212 small blocks1212 large blocks8 tables and 2 chairs: 38 52Used:10 large blocks12 small blocks(2 large blocks left)
1212 small blocks1212 large blocks8 tables and 2 chairs: 38 52Used:10 large blocks12 small blocks(2 large blocks left)I can do it again; change one chair into 2 tables; 10 tables and 1 chairProfit ( 3) x 10 ( 5) x 1 35
1212 small blocks1212 large blocks10 tables and 1 chair: 310 51Used:11 large blocks12 small blocks(1 large block left)
1212 small blocks1212 large blocks10 tables and 1 chair: 310 51Used:11 large blocks12 small blocks(1 large block left)I can do it again; change one chair into 2 tables; 12 tables and 0 chairsProfit ( 3) x 12 ( 5) x 0 36
12 small blocks12 large blocks121212 tables and 0 chairs: 312Profit 36Is this the best?Used:12 large blocks12 small blocks(no blocks left)
Now you try:20 small blocks2012 large blocks12How many tables and chairs? 3 5
Another:25 small blocks2012 large blocks12How many tables and chairs? 3 5
tablesmall 12large 12small 20large 12small 25large 12 3chair rofit605250484644403836Not manysmall blocksNot manysmall blocksMany smallblocks
SummaryM # smallblocksMN # largeblocksNTableChair
Two cases:M # smallblocksMN # largeblocksNTableChair
Two cases:1. Many small blocks:M 2NM # smallblocksMN # largeblocksNTableChair
Two cases:1. Many small blocks:M 2NM # smallblocksMN # largeblocksNÆ make N chairs, 0 tablesTableChair
Two cases:1. Many small blocks:M 2NM # smallblocksMN # largeblocksNÆ make N chairs, 0 tablesTable2. Not many small blocks: M 2NChair
Two cases:1. Many small blocks:M 2NM # smallblocksMN # largeblocksNÆ make N chairs, 0 tablesTable2. Not many small blocks: M 2NÆ mixture of tables and chairs . . . .Chair
Two cases:1. Many small blocks:M 2NM # smallblocksMN # largeblocksNÆ make N chairs, 0 tablesTable2. Not many small blocks: M 2NChairÆ mixture of tables and chairs . . . .What is the magic number of tables and chairs?
Two cases:1. Many small blocks:M 2NM # smallblocksMN # largeblocksNÆ make N chairs, 0 tablesTable2. Not many small blocks: M 2NChairÆ mixture of tables and chairs . . . .What is the magic number of tables and chairs?Let’s make a mathematical model to find out
M # smallblocksMN # largeblocksNTableChair
X # tables builtY # chairs builtOur variablesM # smallblocksMN # largeblocksNTableChair
X # tables builtY # chairs builtHow many can I build?M # smallblocksMN # largeblocksNTableChair
X # tables builtY # chairs builtHow many can I build?X 2Y M; I have only M small blocksX Y N; I have only N large blocksOur equationsM # smallblocksMN # largeblocksNTableChair
X # tables builtY # chairs builtHow many can I build?X 2Y M; I have only M small blocksX Y N; I have only N large blocksOur equations(Note: these are inequalities, notequalities!)M # smallblocksMN # largeblocksNTableChair
X # tables builtY # chairs builtHow many can I build?X 2Y M; I have only M small blocksX Y N; I have only N large blocksM # smallblocksMN # largeblocksNTableChairLet’s plot all possible choices for X and Y for a given M and N,and then we’ll pick the X,Y that gives the greatest profit.
X # tables builtY # chairs builtHow many can I build?X 2Y M; I have only M small blocksX Y N; I have only N large blocksM # smallblocksMN # largeblocksNTableChairLet’s plot all possible choices for X and Y for a given M and N,and then we’ll pick the X,Y that gives the greatest profit.Solving our equations . . .
Intermission: A primer on linear equations
Intermission: A primer on linear equationsA linear equation:aX bY c, or Y mX b,a,b,c,m numbers, X,Y variables
Intermission: A primer on linear equationsA linear equation:aX cY d, or Y mX b,a,b,c,d,m numbers, X,Y variablesYIf we plot all the X,Y that satisfya linear equation, it forms a line:X
Intermission: A primer on linear equationsA linear equation:aX cY d, or Y mX b,a,b,c,d,m numbers, X,Y variablesYIf we plot all the X,Y that satisfya linear equation, it forms a line:risebrunXSlope of a line:riserun m
Intermission: A primer on linear equationsA linear equation:aX cY d, or Y mX b,a,b,c,d,m numbers, X,Y variablesYIf we plot all the X,Y that satisfya linear equation, it forms a line:risebrunXm -2m -1Slope of a line:riserun mm -1/2m 2m 1m 1/2m 0
Back to our problem . . .
Let’s say M 12, N 12X 2Y 12ÆY -(1/2) X 6slope -1/2X Y 12ÆY -(1) X 12slope -1X # tablesY # chairsM # small blocksN # large blocks
Let’s say M 12, N 12X 2Y 12ÆY -(1/2) X 6slope -1/2X Y 12ÆY -(1) X 12slope -1X # tablesY # chairsM # small blocksN # large blocksWe want to find all possible X and Y that satisfy these two equations.
Let’s say M 12, N 12X 2Y 12ÆY -(1/2) X 6slope -1/2X Y 12ÆY -(1) X 12slope -1X # tablesY # chairsM # small blocksN # large blocksWe want to find all possible X and Y that satisfy these two equations.First draw the equality lines;Y126X Y 12X 2Y 12slope -1slope -1/212X
Now the inequalities:X Y 12which side of the line is this region?Y12X Y 12612X
Now the inequalities:X Y 12which side of the line is this region?Let’s check one point: Is (0,0) in this region?Y12X Y 12612X
Now the inequalities:X Y 12which side of the line is this region?Let’s check one point: Is (0,0) in this region?Yes: When X 0 and Y 0 then 0 0 12Y12X Y 12612X
Now the inequalities:X Y 12which side of the line is this region?Let’s check one point: Is (0,0) in this region?Yes: When X 0 and Y 0 then 0 0 12Y12So the allowedvalues of X and Yare below this line(feasible points)X Y 12612X
X 2Y 12which side of the line is this region?Y126X 2Y 1212X
X 2Y 12which side of the line is this region?YRegion is alsobelow the line126X 2Y 1212X
So the allowed region for both inequalities is the common region (the intersection)YOnly X and Y in this region areallowed (for M 12, N 12)126X Y 1212ANDX 2Y 12X
Now, for X and Y in this region, which one gives the highest profit?Profit; P 3X 5YÆY -(3/5)X P/5, slope -3/5Y126X Y 1212ANDX 2Y 12X
Now, for X and Y in this region, which one gives the highest profit?Profit; P 3X 5YÆY -(3/5)X P/5, slope -3/5Let’s plot the profit line P 3X 5YY126X Y 1212ANDX 2Y 12X
Now, for X and Y in this region, which one gives the highest profit?Profit; P 3X 5YÆY -(3/5)X P/5, slope -3/5, Y intercept P/5Along the profit line, all (X,Y) give the same profit PSlope -1YSlope -3/512All X and Yalong thisline givethe sameprofit PSlope -1/2Slope -3/5P/56Slope -1/212X
Now, for X and Y in this region, which one gives the highest profit?Profit; P 3X 5YÆY -(3/5)X P/5, slope -3/5, Y intercept P/5Along the profit line, all (X,Y) give the same profit PSlope -1YSlope -3/512Slope -1/2higher profit6lower profit12X
Now, for X and Y in this region, which one gives the highest profit?Profit; P 3X 5YY12ÆY -(3/5)X P/5, slope -3/5, Y intercept P/5What is the highestprofit for those X,Y inthe feasible region?higher profit6lower profit12X
Now, for X and Y in this region, which one gives the highest profit?Profit; P 3X 5YY12ÆY -(3/5)X P/5, slope -3/5, Y intercept P/5We keep moving theprofit line upwardsuntil the last feasiblepoint is touchedhigher profit6lower profit12X
Now, for X and Y in this region, which one gives the highest profit?Profit; P 3X 5YY12ÆY -(3/5)X P/5, slope -3/5, Y intercept P/5We keep moving theprofit line upwardsuntil the last feasiblepoint is touched6It is this line!12X
Now, for X and Y in this region, which one gives the highest profit?Profit; P 3X 5YY12ÆY -(3/5)X P/5, slope -3/5, Y intercept P/5We keep moving theprofit line upwardsuntil the last feasiblepoint is touched6It is this line!And this is the onlypoint; (12, 0)12X
Now, for X and Y in this region, which one gives the highest profit?Profit; P 3X 5YY12ÆY -(3/5)X P/5, slope -3/5, Y intercept P/5So the greatest profitis achieved whenX 12 and Y 0;Profit 36636 3X 5Y12X
Now let’s look at Case 1: M 2N Æ N M/2X 2Y MÆY -(1/2) X M/2slope -1/2X Y NÆY -(1) X Nslope -1X # tablesY # chairsM # small blocksN # large blocksYM/2NX Y NX 2Y Mslope -1/2slope -1NX
Now let’s look at Case 1: M 2N Æ N M/2X 2Y MÆY -(1/2) X M/2slope -1/2X Y NÆY -(1) X Nslope -1X # tablesY # chairsM # small blocksN # large blocksWe want to find all possible X and Y that satisfy these two equations.YM/2NX Y NX 2Y Mslope -1/2slope -1NX
Now let’s look at Case 1: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksWe want to find all possible X and Y that satisfy these two equations.YM/2NFeasibleregionX Y NX 2Y Mslope -1/2slope -1NX
Now let’s look at Case 1: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksNow plot the profit line: 3X 5Y PYM/2NFeasibleregionX Y NX 2Y Mslope -1/2slope -1NX
Now let’s look at Case 1: M 2N Æ N M/2X 2Y MX Y NNow plot the profit line: 3X 5Y PX # tablesY # chairsM # small blocksN # large blocks(slope -3/5)YM/2NX Y NX 2Y Mslope -1/2profit lineslope -1NX
Now let’s look at Case 1: M 2N Æ N M/2X # tablesY # chairsM # small blocksN # large blocksX 2Y MX Y NMove the profit line up until it last touches the feasible regionYM/2NX Y NX 2Y Mslope -1/2Greatestprofit lineslope -1NX
Now let’s look at Case 1: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksYM/2Point(0,N)NX Y NX 2Y Mslope -1/2Greatestprofit lineslope -1NX
Now let’s look at Case 1: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksSo the greatest profit is achieved when X 0, Y NYM/2Point(0,N)NX Y NX 2Y Mslope -1/2Greatestprofit lineslope -1NX
Now let’s look at Case 1: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksSo the greatest profit is achieved when X 0, Y NYM/2Point(0,N)NMake N chairs0 tablesX Y NX 2Y Mslope -1/2Greatestprofit lineslope -1NX
Now let’s look at Case 2: M 2N Æ N M/2X 2Y MÆY -(1/2) X M/2slope -1/2X Y NÆY -(1) X Nslope -1X # tablesY # chairsM # small blocksN # large blocksYNM/2X Y NX 2Y Mslope -1slope -1/2NX
Now let’s look at Case 2: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksThe feasible region is:YNM/2X Y NX 2Y Mslope -1slope -1/2NX
Now let’s look at Case 2: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksThe feasible region isYNM/2X Y NNote there is a new corner; RX 2Y Mslope -1Rslope -1/2NX
Now let’s look at Case 2: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksNow let’s add the profit linesYNM/2X Y NX 2Y Mslope -1Rslope -1/2NX
Now let’s look at Case 2: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksNow let’s add the profit linesYThe highest one touches R!NM/2X Y NX 2Y MRNX
Now let’s look at Case 2: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksNow let’s add the profit linesYThe highest one touches R!NM/2X Y NX 2Y MSo R is the point (X,Y) thathas the greatest profitRNX
Now let’s look at Case 2: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksWhat is (X,Y) at point R?YNM/2X Y NX 2Y MRNX
Now let’s look at Case 2: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksWhat is (X,Y) at point R?YNM/2It is where the two linesX Y N and X 2Y Mmeet.X Y NX 2Y MRNX
Now let’s look at Case 2: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksWhat is (X,Y) at point R?YNM/2Y M-NIt is where the two linesX Y N and X 2Y Mmeet.X Y NX 2Y MWe find that X 2N - MY M-NRX 2N-MNX
Now let’s look at Case 2: M 2N Æ N M/2X 2Y MX Y NX # tablesY # chairsM # small blocksN # large blocksSo we have our answer:YNM/2Y M-NIf M 2N then we makeX 2N-M tables andY M – N chairsX Y NX 2Y MRX 2N-MNX
table 3chair 3334353336M 20N 12M 644403836M 25N 12M 4644403836M 12N 12M 2NX 2N – M 12Y M–N 0X 2N – M 4Y M–N 8X 0Y N 12
A problem for you . . .
A problem for you . . .hours per kghours availableColumbianMexicanroaster A104roaster B0212grind/package321832.5profit/kg
A problem for you . . .hours per kghours availableColumbianMexicanroaster A104roaster B0212grind/package321832.5profit/kgVariables:XY
X 4roaster A
X 4roaster AYX
X 4roaster AYX 44X
X 4 roaster A2Y 12 roaster BYY 66X 44X
X 4 roaster A2Y 12 roaster B3X 2Y 18 grinding, packagingYY 66X 43X 2Y 18 slope -1.54X
X 4 roaster A2Y 12 roaster B3X 2Y 18 grinding, packagingYY 66X 43X 2Y 18 slope -1.54X
X 4 roaster A2Y 12 roaster B3X 2Y 18 grinding, packagingY6Y 6(2,6)X 4(4,3)3X 2Y 18 slope -1.54X
X 42Y 123X 2Y 183X 2.5Y PY6roaster Aroaster Bgrinding, packagingprofit line, slope -1.2Y 6(2,6)X 4(4,3)3X 2Y 18 slop
powerful mathematical methods used by businesses and companies to solve problems and help them make the best decisions. “Operations Research” is the profession that applies mathematical methods like this to problems arising in industry, healthcare, finance, etc.
Factor each expression or equation, if possible. Solve for x if you are working with an equation. Solve for x, find the roots, find the solution, find zeros, and find x intercept mean the same. 1. Factor 4 20xx32 2. Solve xx2 1 3. Solve xx2 60 4. Solve x2 49 0 5. Solve xx2 2 1 0 6. Solve 2 5 3xx2 7. Factor 0y22 8. Factor 25 16xy22
We create a number bond: Solve the addition equation: 7 1 5 9 Solve the original subtraction equation: 9 2 7 5 Use addition to solve a subtraction equation. 10 2 6 5 Think about addition to solve: 6 1 5 10 Create a number bond: Solve the addition equation: 6 1 5 10 Solve the original subtraction equation: 10 2
IBDP MATHEMATICS: ANALYSIS AND APPROACHES SYLLABUS SL 1.1 11 General SL 1.2 11 Mathematics SL 1.3 11 Mathematics SL 1.4 11 General 11 Mathematics 12 General SL 1.5 11 Mathematics SL 1.6 11 Mathematic12 Specialist SL 1.7 11 Mathematic* Not change of base SL 1.8 11 Mathematics SL 1.9 11 Mathematics AHL 1.10 11 Mathematic* only partially AHL 1.11 Not covered AHL 1.12 11 Mathematics AHL 1.13 12 .
as HSC Year courses: (in increasing order of difficulty) Mathematics General 1 (CEC), Mathematics General 2, Mathematics (‘2 Unit’), Mathematics Extension 1, and Mathematics Extension 2. Students of the two Mathematics General pathways study the preliminary course, Preliminary Mathematics General, followed by either the HSC Mathematics .
2. 3-4 Philosophy of Mathematics 1. Ontology of mathematics 2. Epistemology of mathematics 3. Axiology of mathematics 3. 5-6 The Foundation of Mathematics 1. Ontological foundation of mathematics 2. Epistemological foundation of mathematics 4. 7-8 Ideology of Mathematics Education 1. Industrial Trainer 2. Technological Pragmatics 3.
ns-and-guidance. . credit Learners for the ability to ‘use and apply standard techniques’ (AO1) and/or to ‘solve problems within mathematics and other contexts’ (AO3) an appropriate proportion of the marks for the question/task must be attributed to the corresponding assessment objective(s). AO3 Solve problems within mathematics and .
AFA.1: Apply mathematics to real-life situations; model real-life phenomena using mathematics. AFA.2: Utilize fractions, decimals, percents, and ratios to write and solve a variety of financial problems. AFA.3: Explore and apply functions to model and explain real-life phenomena and to solve com
Θ, solve LP and get B Step to solve mathematical model. 1) Fixed Bat normal admittance, solve LP 1 and obtain Θ 2) Fixed Θ, solve LP2 and get a new B, 3) Fixed Bat new B, solve LP1 again and get new Θ 4) Repeat step 2) - 3), until ΔΘ(difference between two iterations) approaches 0 . i 0. B B 0 . Θ. 0 0 P. Θ 10 Fix B i,Solve LP- Θ. i .
