When coffee is poured into a mug with no lid, its temperature is measured at 170◦F. Three minutes
later its temperature is 165◦F. Also, the surrounding temperature at the coffee shop is set to 68◦F.
a. Write an initial value problem that will describe the change in the temperature of the coffee, T ,
as a function of time t measured in minutes

The problem of describing the temperature change of coffee as it cools over time can be modeled using **Newton's Law of Cooling**. According to Newton's Law of Cooling, the rate at which the temperature of an object changes is proportional to the difference between the object's temperature and the ambient temperature.

Let:
- \( T(t) \) be the temperature of the coffee at time \( t \) (in minutes),
- \( T_{\text{ambient}} = 68^\circ \text{F} \) is the ambient temperature of the coffee shop,
- \( T(0) = 170^\circ \text{F} \) is the initial temperature of the coffee at \( t = 0 \),
- \( \frac{dT}{dt} \) is the rate of change of the coffee's temperature with respect to time.

### Newton's Law of Cooling states:
\[
\frac{dT}{dt} = -k \left( T(t) - T_{\text{ambient}} \right)
\]
where \( k \) is a positive constant that depends on the cooling properties of the system.

### Step 1: Initial Value Problem
The initial value problem (IVP) consists of:
- The differential equation for cooling:
\[
\frac{dT}{dt} = -k (T(t) - 68)
\]
- The initial condition given by the initial temperature of the coffee at \( t = 0 \):
\[
T(0) = 170
\]

Thus, the initial value problem (IVP) is:
\[
\frac{dT}{dt} = -k (T(t) - 68), \quad T(0) = 170
\]

### Step 2: Additional Information
The problem also provides that the temperature of the coffee after 3 minutes is \( T(3) = 165^\circ \text{F} \). This can be used to determine the value of \( k \) after solving the IVP.

Would you like me to solve the differential equation for \( T(t) \) or calculate the constant \( k \) as the next step?

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### 5 Related Questions:
1. How is Newton's Law of Cooling used to model real-life temperature changes?
2. How do we find the constant \( k \) in Newton's Law of Cooling?
3. What happens to the temperature of the coffee over a long period of time?
4. How can we solve first-order differential equations like this one?
5. How does the ambient temperature affect the rate of cooling?

**Tip:** Newton's Law of Cooling applies to many practical scenarios beyond coffee cooling, including body temperature for forensic science or even heating and cooling systems in homes!

When coffee is poured into a mug with no lid, its temperature is measured at 170°F. Three minutes later its temperature is 165°F. Also, the surrounding temperature at the coffee shop is set to 68°F. a. Write an initial value problem that will describe the change in the temperature of the coffee, T, as a function of time t measured in minutes.

Explore how Newton's Law of Cooling is applied to model the temperature change of coffee. Learn to write an initial value problem describing the cooling process, with specific data including the initial temperature of 170°F, ambient temperature of 68°F, and temperature after 3 minutes of 165°F. Suitable for students in Grades 11-12.

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