Question

The heating bill for a homeowner is directly proportional to the rate at which heat is conducted out of the house and into the surroundings. The average temperature inside and outside of a house is measured on different months and recorded in Table 1. \begin{tabular}{|c|c|c|} \hline Month & Temperature ourside ( \(\left.{ }^{\circ} \mathrm{C}\right)\) & Temperature inside \(\left({ }^{\circ} \mathrm{C}\right)\) \\ \hline Now & 8 & 22 \\ \hline Dec & 5 & 25 \\ \hline Jan & 3 & 20 \\ \hline Feb & 13 & 26 \\ \hline \end{tabular} For which month would the homeowner expect to have the largest heating bill? A. November B. December C. January D. February

Short answer

December

Step-by-step solution

- Understand the problem

The heating bill is directly proportional to the rate of heat conduction, which is influenced by the temperature difference between the inside and outside of the house. Therefore, the largest temperature difference will correspond to the largest heating bill.

- Determine the temperature differences

Calculate the temperature difference for each month as follows: November: Temperature inside: 22°C Temperature outside: 8°C Difference: 22 - 8 = 14°C December: Temperature inside: 25°C Temperature outside: 5°C Difference: 25 - 5 = 20°C January: Temperature inside: 20°C Temperature outside: 3°C Difference: 20 - 3 = 17°C February: Temperature inside: 26°C Temperature outside: 13°C Difference: 26 - 13 = 13°C

- Compare the temperature differences

List all the differences: November: 14°C December: 20°C January: 17°C February: 13°C Identify the largest difference.

- Conclusion

The month with the largest temperature difference and thus the highest heating bill is December, with a difference of 20°C.

Key concepts

Heat Conduction

Heat conduction is the process by which thermal energy is transferred from one material to another. This transfer happens due to the temperature difference between the two materials.
According to Fourier's Law, the rate of heat conduction is proportional to the temperature gradient and the thermal conductivity of the material.
A good conductor of heat, like metals, transfers heat efficiently. In contrast, insulators, like wood or rubber, slow down heat transfer.
Understanding heat conduction helps us manage energy use in buildings and other structures.

Temperature Difference

Temperature difference refers to the disparity in temperature between two spaces. This difference drives the flow of heat from the warmer area to the cooler one.
In the context of the exercise, the inside temperature of a house is warmer compared to the outside temperature. The larger the temperature difference, the faster the rate of heat loss.
Calculating the temperature difference helps us estimate the rate at which a house loses heat. This rate impacts heating costs, as more energy is needed to maintain a comfortable indoor temperature when the outside temperature is much lower.

Proportionality in Physics

Proportionality in physics means that one quantity varies directly as another. In the given exercise, the heating bill is directly proportional to the rate of heat conduction.
This implies that if the temperature difference doubles, the rate at which heat is conducted out of the house also doubles, leading to a higher heating bill.
Understanding this concept helps in making informed decisions to improve energy efficiency. For example, better insulation can reduce heat conduction, even if the temperature difference is significant.

Energy Efficiency

Energy efficiency is about using less energy to provide the same level of service. In homes, this can be achieved by minimizing heat loss and optimizing heating systems.
Good insulation, double-glazed windows, and energy-efficient heating systems can significantly reduce the rate of heat conduction. This leads to lower heating bills, despite large temperature differences.
Investing in energy-efficient solutions not only saves money but also reduces environmental impact by lowering energy consumption and greenhouse gas emissions.