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Step-by-Step Solution
Step 1: Identify the Known Quantities
• Initial resistance of the bulb filament, $R_1 = 100\,\Omega$ at $100^\circ \text{C}$.
• Desired resistance, $R_2 = 200\,\Omega$.
• Temperature coefficient of resistance, $\alpha = 0.005 \text{ per }^\circ \text{C}$.
Step 2: Apply the Resistance-Temperature Relation
The resistance at any temperature $T$ (in °C), with reference to a known resistance at a reference temperature $T_0$, can be written as:
$$
R = R_0 \bigl[ 1 + \alpha \, (T - T_0) \bigr].
$$
However, in many problems, if $T_0$ is taken as $0^\circ \text{C}$ for simplicity, we use:
$$
R = R_0 \bigl[ 1 + \alpha \, T \bigr].
$$
In this question, the reference is $100^\circ \text{C}$ for the first given resistance, so we treat that condition carefully.
Step 3: Write Equations for Given Conditions
1. At $100^\circ \text{C}$:
$$
R_1 = R_0 \bigl[ 1 + \alpha \times 100 \bigr] = 100\,\Omega \quad (1)
$$
2. At the unknown temperature $T$ (in °C):
$$
R_2 = R_0 \bigl[ 1 + \alpha \times T \bigr] = 200\,\Omega \quad (2)
$$
Step 4: Form the Ratio of the Two Resistances
Divide equation (2) by equation (1):
$$
\frac{R_2}{R_1}
= \frac{1 + \alpha \, T}{1 + \alpha \times 100}
= \frac{200}{100}
= 2.
$$
Hence,
$$
2 = \frac{1 + \alpha \, T}{1 + 100\,\alpha}.
$$
Step 5: Substitute and Solve for T
Substitute $\alpha = 0.005$:
$$
2 = \frac{1 + 0.005\,T}{1 + 0.005 \times 100}
= \frac{1 + 0.005\,T}{1 + 0.5}.
$$
So,
$$
2 = \frac{1 + 0.005\,T}{1.5}
\quad \Rightarrow \quad
1 + 0.005\,T = 2 \times 1.5 = 3.
$$
$$
0.005\,T = 3 - 1 = 2
\quad \Rightarrow \quad
T = \frac{2}{0.005} = 400^\circ \text{C}.
$$
Step 6: Conclusion
The temperature at which the filament's resistance becomes $200\,\Omega$ is $\boxed{400^\circ \text{C}}$.
