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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 t ) and Jane's mark (which we shall call j ). Tom has 2 more marks than Jane. Therefore,

t = j + 2

Also, both marks add up to 14. Therefore,

t + j = 14

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 t 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

Then,

t = j + 2 = 6 + 2 = 8

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., x .
  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

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?

  1. Let the cost of one ruler be x rand and the cost of one pen be y rand.

  2. 3 x + 2 y = 21 x + y = 8
  3. First solve the second equation for y :

    y = 8 - x

    and substitute the result into the first equation:

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

    therefore

    y = 8 - 5 y = 3
  4.             One ruler costs R 5,00 and one pen costs R 3,00.

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.

  1. Let the price of a chocolate milkshake be x and the price of a fruitshake be y .

    Price number Total
    Fruit y 3 3 y
    Chocolate x 5 5 x
  2. 3 y + 5 x = 78

    y = x + 2

  3. 3 ( x + 2 ) + 5 x = 78 3 x + 6 + 5 x = 78 8 x = 72 x = 9 y = x+2 = 9 + 2 = 11
  4. 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 FunctionsIf a is a positive number such that a>0, then: a^x = a^yif 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 intersectionAlgebraic 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 x 2 - 3 x + 2 = 0 ?
  2. What are the solutions to the equation x 2 + x = 6 ?
  3. In the equation y = 2 x 2 - 5 x - 18 , which is a value of x when y = 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: x 3 - 14 > 14 - x 4
    1. If x R , write the solution in interval notation.
    2. if x Z and x < 51 , write the solution as a set of integers.
  8. Solve for a : 1 - a 2 - 2 - a 3 > 1
  9. Solve for x : x - 1 = 42 x
  10. Solve for x and y : 7 x + 3 y = 13 and 2 x - 3 y = - 4

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2
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