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Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

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Witch of Agnesi

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

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Quick Route

What is the quickest route across a ploughed field when your speed around the edge is greater?

Scientific Curves

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This problem is aimed at interested and enthusiastic students: although we don't recommend it for mainstream classroom use it provides enrichment and will develop mathematical thinking and knowledge.

Like all NRICH problems it has been carefully designed to be full of mathematical interest and as such might be used in a variety of ways: you might suggest that keen students consider the problem in their own time, use it for a maths club at school or use it in class as a vehicle to foster discussion and mathematical thinking. See our Teachers Guide to Getting Started with rich tasks for more suggestions and ideas.
This problem will be of particular interest to students considering a university course containing an element of mathematical physics.