A tungsten filament lamp is marked 60 W, 230 V…

A tungsten filament lamp is marked $60\ \text{W}$, $230\ \text{V}$. These ratings hold when the filament is glowing at its working temperature of $2400\ ^\circ\text{C}$. The resistance of tungsten rises with temperature according to $R = R_0\,[1 + \alpha\,(T - T_0)]$, where $R_0$ is the resistance at $T_0 = 20\ ^\circ\text{C}$ and $\alpha = 4.5 \times 10^{-3}\ ^\circ\text{C}^{-1}$. Treat $\alpha$ as constant over the whole range. The lamp has been off for a long time in a room at $20\ ^\circ\text{C}$. A student switches it on across a supply of $230\ \text{V}$ and wonders why lamps so often fail at the moment of switching on rather than while they glow steadily.

Assume that the supply voltage stays at $230\ \text{V}$ and that the connecting wires have no resistance. Work with the formula as given, and use it for the filament only. The lamp is the only thing connected to the supply, and a filament is a coil of thin tungsten wire that is heated to a bright white glow by the current in it.

1. What current does the lamp draw at its working temperature?
2. What is the resistance of the filament at its working temperature?
3. What is the resistance of the filament at $20\ ^\circ\text{C}$?
4. What current does the lamp draw at the instant of switching on?
5. What power does it draw at that instant?