If `R_(1)` and `R_(2)` are respectively the filament resistances of a 200 watt bulb and 100 watt bulb designed to operate on the same voltage, then

Options

A. `R_(1)` is two times `R_(2)`
B. `R_(2)` is two times `R_(1)`
C. `R_(2)` is four times `R_(1)`
D. `R_(1)` four times `R_(2)`

Answer

To solve the problem, we need to find the relationship between the resistances \( R_1 \) and \( R_2 \) of the 200 watt and 100 watt bulbs, respectively, when both are designed to operate at the same voltage.

### Step-by-step Solution:

1. **Understand the Power Formula**:
The power \( P \) consumed by a resistor can be expressed using the formula:
\[ P = \frac{V^2}{R} \]
where \( V \) is the voltage across the resistor and \( R \) is the resistance.

2. **Set Up the Equations**:
For the 200 watt bulb:
\[ P_1 = 200 \text{ W} = \frac{V^2}{R_1} \]
For the 100 watt bulb:
\[ P_2 = 100 \text{ W} = \frac{V^2}{R_2} \]

3. **Express Resistances in Terms of Power**:
Rearranging the equations gives us:
\[ R_1 = \frac{V^2}{200} \]
\[ R_2 = \frac{V^2}{100} \]

4. **Find the Ratio of Resistances**:
To find the ratio \( \frac{R_2}{R_1} \), we can substitute the expressions for \( R_1 \) and \( R_2 \):
\[ \frac{R_2}{R_1} = \frac{\frac{V^2}{100}}{\frac{V^2}{200}} = \frac{200}{100} = 2 \]

5. **Conclusion**:
From the ratio, we can conclude that:
\[ R_2 = 2R_1 \]
This means the resistance of the 100 watt bulb \( R_2 \) is twice that of the 200 watt bulb \( R_1 \).

### Final Answer:
Thus, the relationship between the filament resistances is:
\[ R_2 = 2R_1 \]