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The problem you are working on asks for a mathematical model based on the verbal statement:

**"The rate of change \( R \) of the temperature \( T \) of an object is directly proportional to the difference between the temperature \( T \) of the object and the temperature \( T_e \) of the environment."**

This is a description of **Newton's Law of Cooling**, which states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the surrounding temperature.

### Correct Model:

\[
\frac{dT}{dt} = -k(T - T_e)
\]

Where:
- \( \frac{dT}{dt} \) is the rate of change of the temperature of the object.
- \( k \) is the constant of proportionality.
- \( T \) is the temperature of the object.
- \( T_e \) is the constant temperature of the environment.
- The negative sign indicates that the temperature of the object decreases when it is hotter than the environment.

### Explanation:
- The phrase "rate of change of the temperature" refers to \( \frac{dT}{dt} \).
- "Directly proportional to" means that we introduce the proportionality constant \( k \).
- The difference between the object's temperature \( T \) and the environmental temperature \( T_e \) gives us \( (T - T_e) \).

The equation you initially used, \( y = kx \), does not account for the fact that the rate of change depends on the temperature difference, hence the mistake.

Would you like more details on how to solve this differential equation?

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Here are 5 questions to consider:
1. What does the negative sign in Newton's Law of Cooling represent?
2. How does the constant \( k \) affect the rate of temperature change?
3. Can you solve this differential equation for \( T(t) \) if given initial conditions?
4. What are other real-world applications of Newton's Law of Cooling?
5. How does this law apply to heating as opposed to cooling?

**Tip:** Always identify which quantities in a word problem represent variables and constants before starting to model the equation.

Find a mathematical model for the verbal statement: The rate of change R of the temperature T of an object is directly proportional to the difference between the temperature T of the object and the temperature Te of the environment.

Learn how to model the rate of change of temperature using Newton's Law of Cooling. This problem demonstrates the proportional relationship between the temperature of an object and the surrounding environment, using differential equations. Suitable for students in Grades 11-12 or college-level.

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