Two electric bulbs , rated for the same voltage , have powers of `200 W and 100 W`, respectively. If their resistances are `r_(1) and r_(2)` , respectively , then

Options

A. ` r_(1) = 2 r_(2)`
B. ` r_(2) = 2 r_(1)`
C. ` r_(2) = 4 r_(1)`
D. ` r_(1) = 4 r_(2)`

Explanation

To solve the problem, we need to find the relationship between the resistances \( r_1 \) and \( r_2 \) of two electric bulbs rated for the same voltage, with powers of 200 W and 100 W respectively.

### Step-by-Step Solution:

1. **Understand the Power Formula**:
The power \( P \) consumed by an electrical device can be expressed in terms of voltage \( V \) and resistance \( R \) using the formula:
\[ P = \frac{V^2}{R} \]
where \( P \) is the power, \( V \) is the voltage, and \( R \) is the resistance.

2. **Set Up the Equations for Each Bulb**:
For the first bulb (200 W):
\[ P_1 = 200 \, \text{W} = \frac{V^2}{r_1} \]
For the second bulb (100 W):
\[ P_2 = 100 \, \text{W} = \frac{V^2}{r_2} \]

3. **Rearrange the Equations**:
From the power equations, we can express the resistances in terms of power and voltage:
\[ r_1 = \frac{V^2}{P_1} = \frac{V^2}{200} \]
\[ r_2 = \frac{V^2}{P_2} = \frac{V^2}{100} \]

4. **Find the Relationship Between \( r_1 \) and \( r_2 \)**:
Now, we can relate \( r_1 \) and \( r_2 \):
\[ \frac{r_2}{r_1} = \frac{\frac{V^2}{100}}{\frac{V^2}{200}} = \frac{200}{100} = 2 \]
This implies:
\[ r_2 = 2r_1 \]

5. **Conclusion**:
The relationship between the resistances of the two bulbs is:
\[ r_2 = 2r_1 \]

### Final Answer:
The resistance \( r_2 \) of the 100 W bulb is twice the resistance \( r_1 \) of the 200 W bulb.

---