Final answer:
The average speed of air in the duct is approximately 11.2 meters per second, based on the volume of the house to be circulated and the dimensions of the air duct.
Step-by-step explanation:
To calculate the average speed of air in the duct, we must first determine the volume of air that must be circulated. This can be found by calculating the volume of the house, which is given by the dimensions of a rectangular solid. The volume V is calculated as width × length × height, hence V = 13.0 m × 20.0 m × 2.75 m = 715 m³. The air duct must carry this volume every 15 minutes, so we calculate the flow rate Q in cubic meters per second as Q = V / t = 715 m³ / (15 min × 60 s/min) = 0.793 m³/s.
The cross-sectional area A of the duct can be found using the diameter d, with the formula for the area of a circle A = π(d/2)^2. This gives us A = π(0.300 m/2)^2 ≈ 0.0707 m³. The average speed v of air then equals the flow rate divided by the cross-sectional area of the duct, v = Q / A. Therefore, v ≈ 0.793 m³/s / 0.0707 m³ ≈ 11.2 m/s. The calculated average speed of air through the duct is 11.2 meters per second.