Surface Environment, Mine Ventilation and Underground Hazards · Mine Ventilation · 2 marks · Multiple choice
The airway is a rectangle topped by a semicircle of radius r. Its full perimeter is20 m and it carries fixed airflow. Which radius minimises mean air velocity?
Minimising velocity Q/A at fixed Q means maximising area. If rectangular height is h, perimeter gives2h+2r+πr=20. Area=2rh+πr²/2=20r−(2+π/2)r². Differentiating gives20−(4+π)r=0, hence r=20/(4+π), A. The negative quadratic coefficient confirms a maximum.
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