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Practical Calculus Problem for the Design of Water Guttering

Please see the attached file for the fully formatted problems.

MAX FLOW

A company is constructing guttering to carry water. The cross section of the guttering is below:

Each side is the same length and the angle between each side and the hosizontal are equal. That is, the cross section is symmetrical about the vertical line through the mid point of its base.

Denote the qidth of the material used to make the guttering as w.
Denote the length of each side as

Denote the ration of as k.
Denote the angle of inclination of the sides as .

If , find the value of k to maximise the volume of water that the guttering can transport.

If ,find the value of to maximise the volume of water that the guttering can transport..

If , find the value of k to maximise the volume of water that the guttering can transport.

NOTES

Use calculus to find all values of all the tasks listed above.
Show all workings and use both graphs and numbers to denote workings along with diagrams of the guttering where necessary.

Solution Summary

Different shapes of guttering are investigated for their maximum flow. The solution includes diagrams.

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