Final answer:
The average density of the heated gases in the hot-air balloon's envelope is found by equating the lifting force the balloon can generate to the buoyant force, using Archimedes' principle and the known weights and volumes, as well as the outside air density.
Step-by-step explanation:
To solve for the average density of the heated gases in the envelope of the balloon, we must use the principles of buoyancy and the overall lifting force that the balloon can generate. The lifting force corresponds to the weight that the balloon can lift, which includes the weight of the balloon fabric, the basket with gear, the propane tanks, and the additional weight of passengers and objects. The combined weights give us the total force that must be equaled by the buoyant force exerted by the surrounding atmosphere on the heated air within the balloon.
The total lifting weight is given by the sum of the fabric (820 N), basket and gear (1600 N), and additional lift for passengers and objects (3800 N), which equals 6220 N. Applying Archimedes' principle, the buoyant force is equal to the weight of the air displaced by the balloon, which can be calculated using the formula Fb = density of the outside air * volume of the balloon * gravitational acceleration.
Given that the gravitational acceleration is approximately 9.81 m/s2 and the volume of the balloon is 2200 m3, we can set up the equation: 6220 N = 1.23 kg/m3 * 2200 m3 * 9.81 m/s2. Solving for the density of the outside air gives a result in N, which must be converted to kg/m3. The density of the heated gases in the envelope is then inferred from the difference in density required to generate the lifting force.