The resistance of hot tungsten filament is about $$10$$ times the cold resistance. What will be the resistance of $$100$$ W and $$200$$ V lamp when not in use?

The resistance of the lamp when it is in use (hot resistance) can be calculated using the power formula:

$$P = \frac{V^2}{R_{hot}}$$

Rearranging the formula to solve for $$R_{hot}$$:

$$R_{hot} = \frac{V^2}{P}$$

Substituting the given values:

$$R_{hot} = \frac{(200)^2}{100}$$

$$R_{hot} = \frac{40000}{100}$$

$$R_{hot} = 400 \ \Omega$$

According to the problem, the resistance of the hot tungsten filament is about 10 times its cold resistance (the resistance when not in use). This gives us the relation:

$$R_{hot} = 10 \times R_{cold}$$

Therefore, the cold resistance $$R_{cold}$$ is:

$$R_{cold} = \frac{R_{hot}}{10}$$

$$R_{cold} = \frac{400}{10}$$

$$R_{cold} = 40 \ \Omega$$