math10001 mathematical workshop mathematical modelling and problem solving
TRANSCRIPT
MATH10001 Mathematical Workshop
Mathematical Modelling and Problem Solving
Traditional view of maths?
● Maths is useless
● The only jobs maths can be used for are accountancy and teaching
● Maths has no link with the real world
The truth is that mathematicians have changed the world,
Radio, digital revolution
Engineering
Computers
Security
Health
Telecommunications Fourier
Fourier Transforms
Joseph Fourier, 1768-1830
Mobile phones, digital radios, MP3 players etc. all use these ideas.
, MP3 players etc all use the sameideas
Internet shopping
Internet transfer protocols based on mathematics
Credit cards and online shopping
– Modern encryption algorithms like RSA use prime number theorems
Scanners
Security screening at the airport
MRI scanners in hospitals
Mathematical modelling in engineering:
eg. Finite element modelling to reduce vibration
Divide car into small
cuboids/tetrahedra.
Treat it a bit like masses
and springs in a network.
Vibrations modelled using eigenvalues of matrices.
Traffic Management
Variable speed signs “shock waves”
Many models of traffic flow Road design Traffic control Better throughput Better safety
Real WorldMathematical
Model
Explanations
& PredictionsSolution
modelling
problem
solving
interpretation
testing Modelling Cycle
Steps in the modelling cycle
1. Identify the problem
2. Define the variables and parameters
• Variables are quantities that can change in a problem
• Parameters are quantities that remain constant
3. Make assumptions
4. Write down a model - relationships between variables
5. Solve the model
6. Test the solution against the real life problem
7. Refine the model
Example – modelling the growth of bacteria
We start with 500 bacteria in a Petri dish. After one day we have 525 bacteria, after two days we have 551 bacteria.
1. The problem is to find a formula for the number of bacteria after n days.
2. Variables include the population, growth rate, time, temperature, amount of food, amount of space left in dish etc. Parameters include initial population, size of dish, initial amount of food.
3. To simplify the problem we make certain assumptions – ignore the amount of food, temperature and space in dish and assume that the growth rate is constant.
4. Define the model: first introduce some notation:
We write down the relationship
5. We have
Assuming a growth rate of 0.05 (from our observations) we get
population initial
days after population
rate growth
experiment of start from days in time
0a
na
k
n
n
.1)1( 111 nakkaaa nnnn any for
.)1(...)1()1( 022
1 akakaka nnnn
.0500)05.1()1( 0 naka nnn all for
6. Test the solution: we test our solution against observations.
We can see that the model works well at the start but after 6 days the model is not accurately predicting the population. This tells us that we need to modify our model.
Day Actual population Predicted population
0 500 500
1 525 525
2 551 551
3 575 579
4 598 608
5 610 638
6 620 670
7. The growth rate appears to be decreasing over time. This could be due to a change in food available or room to grow. We chose to ignore those variables in the original model. As the growth rate is changing we could replace our constant rate by one which is a function of n.
Why do we need mathematical models? Why don’t we simply make lots of observations?