It is known that the temperature in the room is +20^∘ C when the outdoor temperature is -20^∘ C

step 1 As the given situation of the question is that it is known that temperature in the room is plus

It is known that the temperature in the room is +20^∘ C when...

It is known that the temperature in the room is $+20^{\circ} \mathrm{C}$ when the outdoor temperature is $-20^{\circ} \mathrm{C}$ and $+10^{\circ} \mathrm{C}$ when the outdoor temperature is $-40^{\circ} \mathrm{C}$. Then what is the temperature $T$ of the radiator heating the room? (Assuming that radiated by the heater is proportional to the temperature difference with the room.) (A) $40^{\circ} \mathrm{C}$ (B) $60^{\circ} \mathrm{C}$ (C) $30^{\circ} \mathrm{C}$ (D) $20^{\circ} \mathrm{C}$

It is known that the temperature in the room is $+20^{\circ} \mathrm{C}$ when the outdoor temperature is $-20^{\circ} \mathrm{C}$ and $+10^{\circ} \mathrm{C}$ when the outdoor temperature is $-40^{\circ} \mathrm{C}$. Then what is the temperature $T$ of the radiator heating the room? (Assuming that radiated by the heater is proportional to the temperature difference with the room.)
(A) $40^{\circ} \mathrm{C}$
(B) $60^{\circ} \mathrm{C}$
(C) $30^{\circ} \mathrm{C}$
(D) $20^{\circ} \mathrm{C}$

Key Concepts

Steady State Heat Balance

In a thermal system at steady state, the heat delivered by the radiator equals the heat lost from the room. This concept ensures that the room's temperature remains constant over time and allows for setting up balance equations relating the radiator, room, and outdoor temperatures.

Temperature Difference as a Driving Force

Heat transfer often depends on the difference in temperature between two bodies. The greater the difference, the higher the rate of heat transfer. In this context, the radiator emits heat proportionally to the difference between its temperature and the room temperature, and the room loses heat proportionally to the difference between its temperature and the outdoor temperature.

Proportional Relationships and Linear Modeling

When heat transfer is proportional to temperature differences, it can be modeled using linear equations. This approach allows one to relate different temperature conditions mathematically and solve for unknown quantities such as the radiator's temperature by setting up and solving a system of linear equations.

System of Linear Equations

By comparing equilibrium conditions under different outdoor temperatures, one can establish a set of linear equations that express the balance of heat gained and lost. Solving these equations provides insights into the relationships between the radiator, room, and outdoor temperatures, demonstrating a common analytical method in thermal physics.

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