In elementary compressible flow theory (Chap. 9), compressed air will exhaust from a small hole in a tank at the mass flow rate m˙≈Cρ, where ρ is the air density in the tank and C is a constant. If ρ0 is the initial density in a tank of volume V, derive a formula for the density change ρ(t) after the hole is opened. Apply your formula to the following case: a spherical tank of diameter 50 cm, with initial pressure 300 kPa and temperature 100°C, and a hole whose initial exhaust rate is 0.01 kg/s. Find the time required for the tank density to drop by 50 percent.