A pot of coffee is brewed to a temperature of 170 degrees Fahrenheit and is put into a room that is 70 degrees.

After 5 minutes, the temperature of the coffee was 158 degrees.

 

Is someone prefers to drink their coffee at 140 degrees, how long should they wait if their coffee was freshly brewed? (Round to the nearest minute)

To solve this problem, we'll use **Newton's Law of Cooling**. Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (room temperature in this case).

The formula for Newton's Law of Cooling is:

\[
T(t) = T_{\text{room}} + (T_{\text{initial}} - T_{\text{room}}) \cdot e^{-kt}
\]

Where:
- \( T(t) \) is the temperature of the coffee at time \( t \).
- \( T_{\text{room}} \) is the ambient (room) temperature.
- \( T_{\text{initial}} \) is the initial temperature of the coffee.
- \( k \) is a constant that depends on the characteristics of the cooling process.
- \( t \) is the time in minutes.

Given:
- \( T_{\text{room}} = 70^\circ \)F
- \( T_{\text{initial}} = 170^\circ \)F
- After 5 minutes, \( T(5) = 158^\circ \)F

We need to find out how long it will take for the coffee to cool down to \( 140^\circ \)F.

### Step 1: Find the constant \( k \)
Using the data after 5 minutes:

\[
158 = 70 + (170 - 70) \cdot e^{-5k}
\]

Solving for \( k \):

\[
158 = 70 + 100 \cdot e^{-5k}
\]
\[
88 = 100 \cdot e^{-5k}
\]
\[
\frac{88}{100} = e^{-5k}
\]
\[
e^{-5k} = 0.88
\]
\[
-5k = \ln(0.88)
\]
\[
k = -\frac{\ln(0.88)}{5}
\]

### Step 2: Find the time when the temperature is \( 140^\circ \)F

Now, we want to find \( t \) such that \( T(t) = 140^\circ \)F.

\[
140 = 70 + 100 \cdot e^{-kt}
\]

Solving for \( t \):

\[
70 = 100 \cdot e^{-kt}
\]
\[
\frac{70}{100} = e^{-kt}
\]
\[
e^{-kt} = 0.7
\]
\[
-kt = \ln(0.7)
\]
\[
t = \frac{-\ln(0.7)}{k}
\]

Let's compute this using the value of \( k \) we found earlier.

### Calculation

Let's calculate \( k \) and then find the time \( t \).The coffee will take approximately **14 minutes** to cool down to \(140^\circ\) Fahrenheit.

Would you like more details on the calculations or have any other questions?

Here are some related questions you might find interesting:
1. How does the cooling rate change if the room temperature is lower?
2. What if the initial temperature of the coffee is different?
3. Can Newton's Law of Cooling be used for other liquids?
4. How does the coffee cup material affect the cooling time?
5. What if the coffee is left uncovered in the room?

**Tip:** If you prefer your coffee at a specific temperature, using a thermometer might help you time it perfectly!

Learn how to calculate the cooling time for freshly brewed coffee using Newton's Law of Cooling. Given initial and room temperatures, find out how long it takes for coffee to cool from 170°F to 140°F. Step-by-step solution included.

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