Final answer:
To find the average speed of air in the duct, we need to calculate the volume of the house and equate it to the volume of air the duct must carry in 15 minutes. Using the cross-sectional area of the duct, which we find by converting the diameter to radius and then applying the area formula for a circle, we determine the average speed by dividing the house volume by the cross-sectional area and the time in seconds (900 s).
Step-by-step explanation:
To calculate the average speed of air in the duct, we first need to determine the volume of the interior of the house. The volume (V) of a rectangular solid is found by multiplying its length (L), width (W), and height (H), so for the house, V = 13.0 m × 20.0 m × 2.75 m. After calculating the volume of the house, we then have to determine the air volume that the duct can carry in a given time, in this case, every 15 minutes or 900 seconds.
The air duct has a diameter of 0.300 m, so its radius (r) is 0.150 m. The volume (V_duct) of air that the duct can carry is equivalent to the volume of a cylinder, V_duct = πr²h, where π is Pi (approximately 3.14159), r is the radius, and h is the height or length of the cylinder which, in this case, is the distance the air travels in the duct over time. Finally, to find the average speed (v) of air in the duct, we use the formula v = V_duct/h with h being equivalent to the time taken for the air volume to be carried through the duct, in meters per second (m/s).
To solve for the actual numbers, we perform these calculations:
- Calculate the house volume: V_house = 13.0 m × 20.0 m × 2.75 m.
- Determine the air volume the duct must carry in 15 minutes, which is equal to V_house.
- Calculate the cross-sectional area of the duct: A_duct = π×(0.150 m)².
- Using the formula V_duct = V_house and substituting the time of 900 s, calculate v: v = V_duct/900 s.
- Divide this value by the cross-sectional area (A_duct) to find the average speed of the air in the duct.