Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, `100W, 60W and 40W` bulbs have filament resistances `R_(100), R_(60) and R_(40)`, respectively, the relation between these resistances is

Options

A. `1/R_(100) = 1/R_(40) + 1/R_(60)`
B. `R_(100) = R_(40) + R_(60)`
C. `R_(100) gt R_(60) gt R_(40)`
D. `1/R_(100) gt 1/R_(60) gt 1/R_(40)` .

Explanation

To solve the problem, we need to analyze the relationship between the power ratings of the bulbs and their filament resistances. The power \( P \) consumed by an incandescent bulb can be expressed using the formula:

\[ P = \frac{V^2}{R} \]

where \( V \) is the voltage across the bulb and \( R \) is the resistance of the filament.

### Step-by-Step Solution:

1. **Understanding Power and Resistance Relationship**:
From the formula \( P = \frac{V^2}{R} \), we can rearrange it to express resistance in terms of power:
\[ R = \frac{V^2}{P} \]
This shows that resistance \( R \) is inversely proportional to power \( P \) for a constant voltage \( V \).

2. **Setting Up the Resistance for Each Bulb**:
For the three bulbs, we can write:
- For the 100W bulb:
\[ R_{100} = \frac{V^2}{100} \]
- For the 60W bulb:
\[ R_{60} = \frac{V^2}{60} \]
- For the 40W bulb:
\[ R_{40} = \frac{V^2}{40} \]

3. **Comparing the Resistances**:
Since \( V^2 \) is constant for all three bulbs, we can compare the resistances directly:
- From the equations, we can see:
\[ R_{100} = \frac{V^2}{100}, \quad R_{60} = \frac{V^2}{60}, \quad R_{40} = \frac{V^2}{40} \]
- This implies:
\[ \frac{1}{R_{100}} = \frac{100}{V^2}, \quad \frac{1}{R_{60}} = \frac{60}{V^2}, \quad \frac{1}{R_{40}} = \frac{40}{V^2} \]

4. **Establishing the Order of Resistances**:
Since the power ratings are in descending order (100W > 60W > 40W), the resistances will be in ascending order:
\[ R_{100} < R_{60} < R_{40} \]

5. **Final Relation**:
Therefore, we can conclude that:
\[ R_{100} < R_{60} < R_{40} \]

### Conclusion:
The relationship between the resistances of the bulbs is:
\[ R_{100} < R_{60} < R_{40} \]