Question
The main uptake air duct of a forced air gas heater is 0.300 m in diameter. What is the average speed of air in the duct if it carries a volume equal to that of the house’s interior every 15 min? The inside volume of the house is equivalent to a rectangular solid 13.0 m wide by 20.0 m long by 2.75 m high.
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Final Answer

11.2 m/s11.2 \textrm{ m/s}

Solution video

OpenStax College Physics for AP® Courses, Chapter 12, Problem 12 (Problems & Exercises)

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Video Transcript
This is College Physics Answers with Shaun Dychko. The air intake duct of a forced air furnace, we assume is cylindrical with a circular cross-section. And it has a diameter of 0.3 meters, in 15 minutes it will fill up an entire volume of the house of length, 20 meters and width 30 meters and height 2.75 meters. And we convert this minutes into seconds because we usually want our MKS units in our… in our formulas here. Meters kilograms and seconds. So 15 minutes is multiplied by 60 seconds per minute, giving us 900 seconds. So we want to find what speed the air is flowing through this duct. And we know that the volume flow rate of this air is going to be the total volume delivered through the duct and divided by the time it takes to do that. And this also is equal to the cross-sectional area of the duct, multiply by the speed of the air through it. So we divide both sides by area and we get the speed, then the average speed. It's the volume divided by cross-sectional area times time. So the volume that goes to the duct is the volume of the house, we're told, in 15 minutes. And so that volume of this rectangular cuboid is length times, width, times, height, and then the area of the duct is pi times radius squared. And we substitute radius with diameter divided by two. And then we plug all these into our formula for the average speed to get the volume of air, which is volume of the house divided by the cross-sectional area of the duct and multiplied by time. So thatэs 20 meters times 13 meters times two and three quarter meters divided by PI times 0.300 meters diameter divided by two to get the radius squared times 900 seconds. And that is 11.2 meters per second is the speed of the air through the duct.