Answer: 50 minutes
Explanation:
The energy needed to heat air inside the room is the electric energy dissipated by the resistance. It is known after using First Principle of Thermodynamics:
[tex]Q_{in,air} = \dot W_{dis, heater} \cdot \Delta t[/tex]
[tex]\rho_{air} \cdot V_{room} \cdot c_{p,air} \cdot \Delta T = \dot W_{dis,heater} \cdot \Delta t[/tex]
The needed time is:
[tex]\Delta t =\frac{\rho_{air}\cdot V_{room}\cdot c_{p,air} \cdot\Delta T}{\dot W_{dis,heater}}[/tex]
Where [tex]\rho_{air} = 1.20 \frac{kg}{m^{3}}[/tex] and [tex]c_{p,air} = 1.012 \frac{kJ}{kg \cdot ^{\circ} C}[/tex]:
[tex]\Delta t = 3005.640 s (50.094 min)[/tex]
