Step 1: Formulating the Heat Balance Equation

Let the resistance of the heating element be $R_h$, the latent heat required to melt the ice be $Q = mL$, and the rate of heat loss to the surroundings be $r$ (in units of energy per time). Since the ice is at $0^\circ\text{C}$ and the surroundings are at room temperature, heat flows into the vessel from the surroundings only if the vessel is colder, but here the heating element is the source. The text implies a heat loss $r$ *from* the vessel to the surroundings, or a constant term affecting efficiency. Given the context of “heat loss to surrounding,” we write the energy conservation equation as:

$$ \text{Energy Supplied} = \text{Energy to Melt Ice} + \text{Energy Lost} $$ $$ \frac{V^2}{R_h} t = Q + r t $$

Rearranging for power:

$$ \frac{V^2}{R_h} = \frac{Q}{t} + r $$

Let $k = \frac{1}{R_h}$. The equation becomes $V^2 k = \frac{Q}{t} + r$.

Step 2: Analyzing Cases A and B

Case A: $V_A = 380 \, \text{V}$, $t_A = 4 \, \text{min}$.

$$ (380)^2 k = \frac{Q}{4} + r $$ $$ 144400 k = \frac{Q}{4} + r \quad \dots(1) $$

Case B: $V_B = 220 \, \text{V}$, $t_B = 20 \, \text{min}$.

$$ (220)^2 k = \frac{Q}{20} + r $$ $$ 48400 k = \frac{Q}{20} + r \quad \dots(2) $$

Step 3: Solving for Constants

From equation (1), multiply by 4:

$$ Q = 4(144400 k – r) = 577600 k – 4r $$

From equation (2), multiply by 20:

$$ Q = 20(48400 k – r) = 968000 k – 20r $$

Equating the expressions for $Q$:

$$ 577600 k – 4r = 968000 k – 20r $$ $$ 16r = 390400 k $$ $$ r = 24400 k $$

Step 4: Analyzing Case C

Case C: $V_C = 110 \, \text{V}$. We need to find $t_C$.

The power supplied by the heater is:

$$ P_{\text{supplied}} = (110)^2 k = 12100 k $$

The rate of heat loss to the surroundings is:

$$ P_{\text{loss}} = r = 24400 k $$

Comparing the power supplied to the heat loss:

$$ P_{\text{supplied}} (12100 k) < P_{\text{loss}} (24400 k) $$

The heating element supplies heat at a rate lower than the rate at which heat is lost to the surroundings. Therefore, the net heat available to melt the ice is negative. The system will never reach the energy required to melt the ice.