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To determine the recovery time of a hot water heater, we first need to calculate the thermal energy required to heat the water. Thermal energy refers to the energy that is transferred to raise the water's temperature. In this scenario, we use the formula:\[ Q = mc\Delta T \]where

• \( Q \) is the thermal energy in joules (J),
• \( m \) is the mass of the water in kilograms (kg),
• \( c \) is the specific heat capacity of the water (more on this in the next section), and
• \( \Delta T \) is the change in temperature in degrees Celsius (°C).

Calculate the change in temperature by subtracting the initial temperature (\( 11^{\circ} \mathrm{C} \)) from the desired temperature (\( 53^{\circ} \mathrm{C} \)), resulting in a \( \Delta T \) of \( 42^{\circ} \mathrm{C} \). We already know the mass and specific heat capacity. Given this information, you can find the thermal energy required.

Specific heat capacity is a crucial concept when it comes to heating substances, especially water. Specific heat capacity is the amount of heat energy required to raise the temperature of one kilogram of a substance by one degree Celsius. For water, this value is relatively high at \( 4186 \, \mathrm{J/kg^{\bullet}C} \), meaning water requires a significant amount of energy to change its temperature.
This property makes water an excellent substance for maintaining temperatures in heating systems. Since the specific heat capacity of water is consistent, you can reliably calculate the energy needed to achieve a desired temperature change.In thermal energy calculations, always ensure to measure or know the substance's specific heat capacity you are working with to make accurate predictions or solutions.

Electric resistance heating involves an electric current passing through a resistance, converting electrical energy into heat. This principle is used in electric water heaters to elevate water to a desired temperature. The efficiency and power of this heating system depend upon the voltage supplied and the resistance of the heating element.To assess the power supplied by the heater, you can apply the formula:\[ P = \frac{V^2}{R} \]where

• \( P \) is the power in watts (W),
• \( V \) is the voltage in volts (V),
• \( R \) is the resistance in ohms (\( \Omega \)).

In the given exercise, with a voltage of \( 120 \, \mathrm{V} \) and a resistance of \( 3.0 \, \Omega \), the power is determined to be \( 4800 \, \mathrm{W} \). Larger power output implies quicker heating; thus, a factor that can significantly influence recovery time. Always consider these parameters early to ensure efficient system design and operation.