Final answer:
The average density of the heated air inside the envelope of the hot air balloon is 0.9585 kg/m³. This is found by using the principles of buoyancy and Archimedes' Principle and the information provided about the total lifting force.
Step-by-step explanation:
To calculate the average density of the heated air inside the envelope of the hot air balloon, we first need to apply the principles of buoyancy and Archimedes' Principle. From the given information, the balloon can lift a total force including its own weight and the propane tanks, which is the sum of the fabric weight, the basket weight, and the additional weight of passengers and contents:
900 N (fabric) + 1700 N (basket and propane) + 3200 N (passengers and contents) = 5800 N (total lifting force).
The buoyancy force (Fb) is the weight of the air displaced by the balloon, which equals the force that the balloon can lift (5800 N). The weight of the air displaced can be calculated as the volume of the balloon multiplied by the density of the outside air and the acceleration due to gravity (g = 9.8 m/s²):
Fb = (Volume of the balloon) x (Density of the outside air) x g = 2200 m³ x 1.23 kg/m³ x 9.8 m/s² = 26449.4 N.
The weight of the hot air inside the balloon (Fg) is then the difference between the buoyancy force and the total lifting force of the balloon:
Fg = Fb - Total lifting force = 26449.4 N - 5800 N = 20649.4 N.
To find the density of the hot air inside the balloon, we divide the Fg by the volume of the balloon and the acceleration due to gravity:
Density = Fg / (Volume x g) = 20649.4 N / (2200 m³ x 9.8 m/s²) = 0.9585 kg/m³