Newton's Law of Cooling describes how the temperature of an object changes over time when it's exposed to a different ambient temperature. The rate of change of the object's temperature is proportional to the difference between its own temperature and the ambient temperature. Essentially, it tells us how quickly something heats up or cools down!
Mathematically, it's expressed as: $\frac{dT}{dt} = k(T - T_a)$, where $T$ is the temperature of the object at time $t$, $T_a$ is the ambient temperature, and $k$ is a constant that depends on the properties of the object.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Ambient Temperature | a. The constant that depends on the object's properties. |
| 2. Rate of Cooling | b. The temperature of the surrounding environment. |
| 3. Differential Equation | c. The change in temperature with respect to time. |
| 4. Constant of Proportionality (k) | d. An equation involving derivatives. |
| 5. Initial Condition | e. The temperature of an object at time t=0. |
Newton's Law of Cooling states that the rate of change of the temperature of an object is __________ to the difference between its own __________ and the __________ temperature. The constant of __________ depends on the __________ of the object and its surface area.
Imagine you have a cup of coffee at 90°C in a room that is 20°C. How would changing the constant $k$ (e.g., by using a different cup) affect how quickly the coffee cools down? Explain your reasoning.
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