What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

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?

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?

How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.

How do you choose your planting levels to minimise the total loss at harvest time?

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

Why MUST these statistical statements probably be at least a little bit wrong?

This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.

Invent scenarios which would give rise to these probability density functions.

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Are these statistical statements sometimes, always or never true? Or it is impossible to say?

Which line graph, equations and physical processes go together?

Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

Can you sketch these difficult curves, which have uses in mathematical modelling?

Can you construct a cubic equation with a certain distance between its turning points?

Can you match the charts of these functions to the charts of their integrals?

Here are several equations from real life. Can you work out which measurements are possible from each equation?

Explore the properties of matrix transformations with these 10 stimulating questions.

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.

Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation

When you change the units, do the numbers get bigger or smaller?

Work with numbers big and small to estimate and calculate various quantities in biological contexts.

Looking at small values of functions. Motivating the existence of the Taylor expansion.

Build up the concept of the Taylor series

Work out the numerical values for these physical quantities.

Estimate these curious quantities sufficiently accurately that you can rank them in order of size

Use vectors and matrices to explore the symmetries of crystals.

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

See how enormously large quantities can cancel out to give a good approximation to the factorial function.

Can you make matrices which will fix one lucky vector and crush another to zero?

Explore the meaning of the scalar and vector cross products and see how the two are related.

This problem explores the biology behind Rudolph's glowing red nose.

Match the descriptions of physical processes to these differential equations.

Which of these infinitely deep vessels will eventually full up?

Get further into power series using the fascinating Bessel's equation.

Which units would you choose best to fit these situations?

Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

Formulate and investigate a simple mathematical model for the design of a table mat.

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.