This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.
The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?
Why MUST these statistical statements probably be at least a little bit wrong?
See how enormously large quantities can cancel out to give a good approximation to the factorial function.
Invent scenarios which would give rise to these probability density functions.
Which line graph, equations and physical processes go together?
Are these statistical statements sometimes, always or never true? Or it is impossible to say?
Work with numbers big and small to estimate and calculate various quantities in physical contexts.
Get further into power series using the fascinating Bessel's equation.
Here are several equations from real life. Can you work out which measurements are possible from each equation?
Use vectors and matrices to explore the symmetries of crystals.
Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?
Get some practice using big and small numbers in chemistry.
Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.
By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.
Work out the numerical values for these physical quantities.
How much energy has gone into warming the planet?
Work with numbers big and small to estimate and calculate various quantities in biological contexts.
Build up the concept of the Taylor series
Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?
Can you sketch these difficult curves, which have uses in mathematical modelling?
How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?
Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?
Formulate and investigate a simple mathematical model for the design of a table mat.
Was it possible that this dangerous driving penalty was issued in error?
Can you make matrices which will fix one lucky vector and crush another to zero?
Which pdfs match the curves?
Estimate these curious quantities sufficiently accurately that you can rank them in order of size
Starting with two basic vector steps, which destinations can you reach on a vector walk?
Can you find the volumes of the mathematical vessels?
Go on a vector walk and determine which points on the walk are closest to the origin.
Explore the meaning of the scalar and vector cross products and see how the two are related.
How would you go about estimating populations of dolphins?
In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.
When you change the units, do the numbers get bigger or smaller?
Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation
Work with numbers big and small to estimate and calulate various quantities in biological contexts.
Who will be the first investor to pay off their debt?
Match the descriptions of physical processes to these differential equations.
Explore the properties of matrix transformations with these 10 stimulating questions.
Which units would you choose best to fit these situations?
Explore the shape of a square after it is transformed by the action of a matrix.
Look at the advanced way of viewing sin and cos through their power series.
Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.
Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?
To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...
Make an accurate diagram of the solar system and explore the concept of a grand conjunction.
Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.
Explore the relationship between resistance and temperature
Can you work out what this procedure is doing?