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?

---

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!