Final answer:
The average speed of air in the duct is calculated by dividing the volume of the house by the product of the cross-sectional area of the duct and the time interval. After performing the necessary calculations, we find that the average speed is approximately 3.4 m/s, which is option (b).
Step-by-step explanation:
To calculate the average speed of air in the duct, we first need to find the volume of the house's interior and then how much air needs to pass through the duct every second to replace this volume in 15 minutes. The volume of the house is calculated by multiplying the width, length, and height of the rectangular solid representing the house. That is, Volume = 13.0 m × 20.0 m × 2.75 m = 715 m³.
The air duct is a cylinder, and its volume flow rate is the volume of air passing through per unit time. Given the diameter of the duct is 0.300 m, the radius is half of that, which is 0.150 m.
Using the formula for the volume of a cylinder, πr²h, where r is the radius and h is the height (or length in this context), we can express the volume of the cylinder as π × (0.150m) ² × h. In this case, h would be the length equivalent to speed × time since we're looking for the speed of air (which is the distance covered per unit time).
Since we want the entire air in the house to be replaced every 15 minutes (or 900 seconds), we can set up the equation as follows:
Volume of house / time = π × (0.150m)² × (speed), which simplifies to
715 m³ / 900 s = π × (0.150m)² × (speed).
After solving for speed, we get:
Speed = (715 m³ / 900 s) / (π × (0.150m)²) ≈ 3.4 m/s
So, the average speed of air in the duct is approximately 3.4 m/s, which corresponds to option (b).