If a thermometer is taken outdoors where the temperature is 0 degree Celsius, from a room in which the temperature is 21 degree Celsius and the reading drops to 10 degree Celsius in 1 minute, how long after its removal will the reading be 5 degree Celsius.

To solve this problem, we can use **Newton's Law of Cooling**, which is given by:

\[
T(t) = T_{\text{env}} + (T_0 - T_{\text{env}}) e^{-kt}
\]

Where:
- \( T(t) \) is the temperature of the object at time \( t \).
- \( T_{\text{env}} \) is the ambient temperature (0°C in this case).
- \( T_0 \) is the initial temperature of the object (21°C).
- \( k \) is the cooling constant.
- \( t \) is the time elapsed.

### Step 1: Determine the cooling constant \( k \)
First, we need to find \( k \) using the information given:

At \( t = 1 \) minute, the temperature drops to 10°C:

\[
10 = 0 + (21 - 0) e^{-k \cdot 1}
\]

\[
10 = 21 e^{-k}
\]

\[
\frac{10}{21} = e^{-k}
\]

Taking the natural logarithm on both sides:

\[
\ln \left(\frac{10}{21}\right) = -k
\]

\[
k = -\ln \left(\frac{10}{21}\right)
\]

### Step 2: Determine the time \( t \) when the temperature is 5°C
Now, we need to find the time \( t \) when the temperature is 5°C:

\[
5 = 0 + (21 - 0) e^{-kt}
\]

\[
\frac{5}{21} = e^{-kt}
\]

Taking the natural logarithm on both sides:

\[
\ln \left(\frac{5}{21}\right) = -kt
\]

Substituting the value of \( k \) we found earlier:

\[
\ln \left(\frac{5}{21}\right) = -\left(-\ln \left(\frac{10}{21}\right)\right)t
\]

\[
t = \frac{\ln \left(\frac{5}{21}\right)}{-\ln \left(\frac{10}{21}\right)}
\]

### Step 3: Calculate the value of \( t \)

Let's compute this value.

\[
t = \frac{\ln \left(\frac{5}{21}\right)}{-\ln \left(\frac{10}{21}\right)}
\]

\[
t = \frac{\ln \left(\frac{5}{21}\right)}{\ln \left(\frac{21}{10}\right)}
\]

Calculating the logarithms:

\[
t \approx \frac{\ln(0.2381)}{\ln(2.1)} \approx \frac{-1.433}{0.741}
\]

\[
t \approx 1.934 \text{ minutes}
\]

### Final Answer
It will take approximately **1.934 minutes** after removal for the thermometer reading to be 5°C.

Would you like further details or have any questions? 

Here are 5 related questions for practice:
1. What is Newton's Law of Cooling and when is it applied?
2. How do you determine the cooling constant \( k \) from experimental data?
3. How does changing the ambient temperature affect the cooling time?
4. What would happen if the thermometer was taken to an environment with a temperature below 0°C?
5. Can Newton's Law of Cooling be used to estimate the time of death in forensic science?

**Tip:** Always take the natural logarithm of both sides when solving exponential equations for time, as it helps isolate the variable of interest.

Explore how to solve a thermodynamics problem using Newton's Law of Cooling. Learn about exponential decay and the application of this principle in calculating temperature changes. Suitable for high school students.

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