In the pastime of hot-air ballooning, the air inside a balloon—called the envelope—is heated until its density is lower than that of the surrounding atmosphere, allowing the envelope and an attached basket to be lifted into the air. One team is preparing its balloon for a flight on a dry day at 16°C. The envelope has a volume of 3125 m³ when inflated and, excluding the mass of the air, the total mass of the envelope and the basket loaded with fuel, heater, passengers, and sandwiches, is 586 kg.

(a) What would be the mass of air in the envelope if it were at 16 °C?

(b) To what temperature must the air in the envelope be heated to lift it and the loaded basket?

(c) What is the mass of the air inside the envelope at that temperature? Assume that the volume of the loaded basket is negligible and that the molar mass of the air is 28.97 g · mol^–1.