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# Mathematical models

## Mathematical Models

### Introduction

Tom and Jane are friends. Tom picked up Jane's Physics test paper, but will not tell Jane what her marks are. He knows that Jane hates maths so he decided to tease her. Tom says: “I have 2 marks more than you do and the sum of both our marks is equal to 14. How much did we get?”

Let's help Jane find out what her marks are. We have two unknowns, Tom's mark (which we shall call tt) and Jane's mark (which we shall call jj). Tom has 2 more marks than Jane. Therefore,

t = j + 2 t = j + 2
(1)

Also, both marks add up to 14. Therefore,

t + j = 14 t + j = 14
(2)

The two equations make up a set of linear (because the highest power is one) simultaneous equations, which we know how to solve! Substitute for tt in the second equation to get:

t + j = 14 j + 2 + j = 14 2 j + 2 = 14 2 ( j + 1 ) = 14 j + 1 = 7 j = 7 - 1 = 6 t + j = 14 j + 2 + j = 14 2 j + 2 = 14 2 ( j + 1 ) = 14 j + 1 = 7 j = 7 - 1 = 6
(3)

Then,

t = j + 2 = 6 + 2 = 8 t = j + 2 = 6 + 2 = 8
(4)

So, we see that Tom scored 8 on his test and Jane scored 6.

This problem is an example of a simple mathematical model. We took a problem and we were able to write a set of equations that represented the problem mathematically. The solution of the equations then gave the solution to the problem.

### Problem Solving Strategy

The purpose of this section is to teach you the skills that you need to be able to take a problem and formulate it mathematically in order to solve it. The general steps to follow are:

1. Read ALL of it !
2. Find out what is requested.
3. Use a variable(s) to denote the unknown quantity/quantities that has/have been requested e.g., xx.
4. Rewrite the information given in terms of the variable(s). That is, translate the words into algebraic expressions.
5. Set up an equation or set of equations (i.e. a mathematical sentence or model) to solve the required variable.
6. Solve the equation algebraically to find the result.

### Application of Mathematical Modelling

#### Exercise 1: Mathematical Modelling: Two variables

Three rulers and two pens have a total cost of R 21,00. One ruler and one pen have a total cost of R 8,00. How much does a ruler costs on its own and how much does a pen cost on its own?

##### Solution
1. Step 1. Translate the problem using variables :

Let the cost of one ruler be xx rand and the cost of one pen be yy rand.

2. Step 2. Rewrite the information in terms of the variables :
3 x + 2 y = 21 x + y = 8 3 x + 2 y = 21 x + y = 8
(5)
3. Step 3. Solve the equations simultaneously :

First solve the second equation for yy:

y = 8 - x y = 8 - x
(6)

and substitute the result into the first equation:

3 x + 2 ( 8 - x ) = 21 3 x + 16 - 2 x = 21 x = 5 3 x + 2 ( 8 - x ) = 21 3 x + 16 - 2 x = 21 x = 5
(7)

therefore

y = 8 - 5 y = 3 y = 8 - 5 y = 3
(8)
4. Step 4. Present the final answers :

One ruler costs R 5,00 and one pen costs R 3,00.

#### Exercise 2: Mathematical Modelling: One variable

A fruit shake costs R2,00 more than a chocolate milkshake. If three fruit shakes and 5 chocolate milkshakes cost R78,00, determine the individual prices.

##### Solution
1. Step 1. Summarise the information in a table :

Let the price of a chocolate milkshake be xx and the price of a fruitshake be yy.

 Price number Total Fruit y y 3 3 y 3 y Chocolate x x 5 5 x 5 x
2. Step 2. Set up a pair of algebraic equations :
3 y + 5 x = 78 3 y + 5 x = 78
(9)

y=x+2y=x+2

3. Step 3. Solve the equations :
3 ( x + 2 ) + 5 x = 78 3 x + 6 + 5 x = 78 8 x = 72 x = 9 y = x+2 = 9 + 2 = 11 3 ( x + 2 ) + 5 x = 78 3 x + 6 + 5 x = 78 8 x = 72 x = 9 y = x+2 = 9 + 2 = 11
(10)
4. Step 4. Present the final answer :

One chocolate milkshake costs R 9,00 and one Fruitshake costs R 11,00

#### Mathematical Models

1. Stephen has 1 l of a mixture containing 69% of salt. How much water must Stephen add to make the mixture 50% salt? Write your answer as a fraction of a litre.

2. The diagonal of a rectangle is 25 cm more than its width. The length of the rectangle is 17 cm more than its width. What are the dimensions of the rectangle?

3. The sum of 27 and 12 is 73 more than an unknown number. Find the unknown number.

4. The two smaller angles in a right-angled triangle are in the ratio of 1:2. What are the sizes of the two angles?

5. George owns a bakery that specialises in wedding cakes. For each wedding cake, it costs George R150 for ingredients, R50 for overhead, and R5 for advertising. George's wedding cakes cost R400 each. As a percentage of George's costs, how much profit does he make for each cake sold?

6. If 4 times a number is increased by 7, the result is 15 less than the square of the number. Find the numbers that satisfy this statement, by formulating an equation and then solving it.

7. The length of a rectangle is 2 cm more than the width of the rectangle. The perimeter of the rectangle is 20 cm. Find the length and the width of the rectangle.


### Summary

• Linear equations A linear equation is an equation where the power of the variable (letter, e.g. x) is 1(one). Has at most one solution
• Quadratic equations A quadratic equation is an equation where the power of the variable is at most 2. Has at most two solutions
• Exponential equations Exponential equations generally have the unknown variable as the power. ka^(x+p) = m Equality for Exponential Functions If a is a positive number such that a > 0, then: a^x = a^y if and only if: x=y
• Linear inequalities A linear inequality is similar to a linear equation and has the power of the variable equal to 1. When you divide or multiply both sides of an inequality by any number with a minus sign, the direction of the inequality changes. Solve as for linear equations
• Linear simultaneous equations When two unknown variables need to be solved for, two equations are required and these equations are known as simultaneous equations. Graphical or algebraic solutions Graphical solution: Draw the graph of each equation and the solution is the co-ordinates of intersection Algebraic solution: Solve equation one, for variable one and then substitute it into equation two.
• Mathematical models Take a problem, write equations that represent it, solve the equations and that solves the problem.

### End of Chapter Exercises

1. What are the roots of the quadratic equation x2-3x+2=0x2-3x+2=0

?

2. What are the solutions to the equation x2+x=6x2+x=6

?

3. In the equation y=2x2-5x-18y=2x2-5x-18, which is a value of xx when y=0y=0

?

4. Manuel has 5 more CDs than Pedro has. Bob has twice as many CDs as Manuel has. Altogether the boys have 63 CDs. Find how many CDs each person has.

5. Seven-eighths of a certain number is 5 more than one-third of the number. Find the number.

6. A man runs to a telephone and back in 15 minutes. His speed on the way to the telephone is 5 m/s and his speed on the way back is 4 m/s. Find the distance to the telephone.

7. Solve the inequality and then answer the questions: x3-14>14-x4x3-14>14-x4
1. If xRxR, write the solution in interval notation.
2. if xZxZ and x<51x<51, write the solution as a set of integers.

8. Solve for aa: 1-a2-2-a3>11-a2-2-a3>1

9. Solve for xx: x-1=42xx-1=42x

10. Solve for xx and yy: 7x+3y=137x+3y=13 and 2x-3y=-42x-3y=-4


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