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## 'Gutter' printed from http://nrich.maths.org/

Some learners may access this general problem more successfully if
a particular cross-section (profile) type is specified at the
start: a simple rectangle for example. This could lead into
consideration of a trapezium (removing the constraint of vertical
sides) .

Algebra could feature at several points in this problem. For
example when learners are :

- identifying useful variables;
- working with a simple rectanglular profile,several depth values
might be used,producing different profile areas which could be
plotted;
- utilising a spreadsheet, deriving a general formula for
calculating the profile area based on depth.

Interpretation of the calculated results is needed and
involves considering the proportions between lengths and and the
implications of that for the shape of the profile.

There is an interesting general principle underlying this
problem: that if the profile is based on a regular polygon the
optimum cross-section is half of that polygon.

Unrushed discussion of why this is the case might lead to
connections being made with other optimisation problems related to
area or volume.

Familiarity with mathematics may lead us instinctively towards
a circular profile as the optimum solution (i.e. a pipe). But that
is not in fact anywhere near the best form, and the experience of
challenging this misconception can be a pathway into new
understanding.