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.

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?

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.

Can you draw the height-time chart as this complicated vessel fills with water?

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

Which of these infinitely deep vessels will eventually full up?

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

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

Was it possible that this dangerous driving penalty was issued in error?

How would you go about estimating populations of dolphins?

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

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

Match the charts of these functions to the charts of their integrals.

Various solids are lowered into a beaker of water. How does the water level rise in each case?

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

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

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

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

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

Use vectors and matrices to explore the symmetries of crystals.

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?

Analyse these beautiful biological images and attempt to rank them in size order.

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

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

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

Match the descriptions of physical processes to these differential equations.

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

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

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

Explore the relationship between resistance and temperature

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

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

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

Build up the concept of the Taylor series

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

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

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

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

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