# Gutter

Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?

A plastic gutter is designed to catch water at the edge of a roof.

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Manufacturers need to minimise the amount of material used to make their product while maximising the volume of water that can be drained.

**What is the optimal cross-section for a gutter?**

You might want to start by investigating gutters with a rectangular cross-section. Choose a fixed length for the cross section and vary the length of the base of the gutter. How does the area of the cross section change?

**Using the same length**, investigate triangular cross-sections. Vary the angle. How does the area of the cross section change?

Finally,

**use the same length**and investigate cross-sections made from circular arcs. Vary the angle that defines the arc. How does the area of the cross section change?

Pick one of the cross-sections - perhaps the rectangle.

In this challenge, why does it help to think about the cross-section (profile) and not at all about how long the gutter might be?

The material "length" of the profile of the rectangle wasn't specified in the problem - perhaps it doesn't matter or perhaps we need to give it a value and see what happens.

By the end you will have used the idea that this problem is about shape and not about size.

What did your answers tell you about the "best shape" for the profile?

The picture suggests two other shapes to work with, but there are other possibilities, which did you think to try? Did you include a trapezium?

We are still waiting for some good student contributions on this problem but here are some thoughts to help you :

You could start with the three shape posibilities in this illustration :

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First the triangle : suppose a length of say 20 cm was folded in the centre, for which angle would the cross-section area be greatest ? If the length had been 40 cm instead of 20 cm would that change your answer ?

Now the rectangle cross-section : maybe start with an overall length of 20 cm again and try different proportions of side length to base. What proportion created the largest area, was that a surprise or an expected result ?

Finally, for now, the arc : for a specified length, say 20 cm, you could curve it a lot or a little. If you curve it a lot you will have more of the full circle but it's a smaller circle. If you curve the material only a little you get less of a larger circle. A spreadsheet might help you find the best choice and ease the burden of calculation.

I hope you enjoy finding that result, if you didn't guess it. Now compare all three results : notice anything ? Can you account for that ?

*This printable worksheet may be useful: Guttter.*

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.

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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.