Calculating Heater Power Consumption at Reduced Voltage
The problem asks us to find the new power consumed by a room heater when the supply voltage decreases, given its original rating. We need to use the relationship between power, voltage, and resistance.
Heater Resistance Calculation
First, we determine the resistance of the heater, which remains constant. The rated power ($P_{rated}$) is $400 \text{ W}$ and the rated voltage ($V_{rated}$) is $220 \text{ V}$. The formula relating power, voltage, and resistance is:
$ P = \frac{V^2}{R} $
We can rearrange this to find the resistance ($R$):
$ R = \frac{V_{rated}^2}{P_{rated}} $
Substituting the given values:
$ R = \frac{(220 \text{ V})^2}{400 \text{ W}} = \frac{48400}{400} \text{ } \Omega = 121 \text{ } \Omega $
So, the resistance of the heater is $121 \text{ } \Omega$.
New Power Consumption
Now, the supply voltage drops to $V_{new} = 200 \text{ V}$. We use the same resistance ($R = 121 \text{ } \Omega$) and the new voltage to calculate the new power consumed ($P_{new}$):
$ P_{new} = \frac{V_{new}^2}{R} $
Substituting the values:
$ P_{new} = \frac{(200 \text{ V})^2}{121 \text{ } \Omega} = \frac{40000}{121} \text{ W} $
$ P_{new} \approx 330.58 \text{ W} $
Rounding this to the nearest watt, the approximate power consumed is $331 \text{ W}$.
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