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

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.

Where should runners start the 200m race so that they have all run the same distance by the finish?

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

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

Look at the advanced way of viewing sin and cos through their power series.

How would you go about estimating populations of dolphins?

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

Build up the concept of the Taylor series

How would you design the tiering of seats in a stadium so that all spectators have a good view?

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

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.

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.

Use vectors and matrices to explore the symmetries of crystals.

Go on a vector walk and determine which points on the walk are closest to the origin.

Explore the shape of a square after it is transformed by the action of a matrix.

Use trigonometry to determine whether solar eclipses on earth can be perfect.

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

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

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

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

Starting with two basic vector steps, which destinations can you reach on a vector walk?

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

Is it really greener to go on the bus, or to buy local?

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

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

Could nanotechnology be used to see if an artery is blocked? Or is this just science fiction?

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

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

Work out the numerical values for these physical quantities.

Make an accurate diagram of the solar system and explore the concept of a grand conjunction.

Get some practice using big and small numbers in chemistry.

In which Olympic event does a human travel fastest? Decide which events to include in your Alternative Record Book.

Can you work out which processes are represented by the graphs?

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

What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?

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

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

Can Jo make a gym bag for her trainers from the piece of fabric she has?

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

Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?