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The problem in the image involves a thermometer taken from a room at 21°C to the outdoors where the temperature is 5°C. After one minute, the thermometer reads 17°C. We are tasked with answering the following questions:

**(a)** What will the thermometer read after 3 more minutes?

**(b)** When will the thermometer read 6°C?

This type of problem is commonly modeled using **Newton's Law of Cooling**, which states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature. The formula is:

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

Where:
- \( T(t) \) is the temperature at time \( t \).
- \( T_{\text{env}} \) is the ambient temperature (outdoors) which is 5°C.
- \( T_0 \) is the initial temperature (21°C in this case).
- \( k \) is a constant that depends on the cooling rate.
- \( t \) is the time in minutes.

Let's break it down:

### 1. Determine the cooling constant \( k \) using the given data after 1 minute.

At \( t = 1 \), the temperature \( T(1) = 17^\circ C \). Substituting the known values into the equation:

\[
17 = 5 + (21 - 5) e^{-k(1)}
\]

We can solve this equation for \( k \).

### 2. Use the value of \( k \) to find the temperature after 3 more minutes (4 minutes in total).

Once we have \( k \), we can substitute \( t = 4 \) into the equation to find the temperature after 4 minutes.

### 3. Solve for the time when the thermometer will read 6°C.

We will set \( T(t) = 6^\circ C \) and solve for \( t \).

I will now calculate these values.Here are the results based on Newton's Law of Cooling:

**(a)** After 3 more minutes (4 minutes total), the temperature on the thermometer will be approximately **10.06°C**.

**(b)** The thermometer will read **6°C** approximately **9.64 minutes** after it was taken outdoors.

Would you like more details or clarifications?

Here are 5 related questions to consider:
1. What assumptions are made in Newton's Law of Cooling?
2. How would the solution change if the thermometer's initial temperature was higher?
3. How is the cooling constant \( k \) affected by different environments?
4. How could we experimentally determine the value of \( k \) in real-world scenarios?
5. How would the time to reach a specific temperature change if the outdoor temperature was lower?

**Tip:** Newton's Law of Cooling assumes that the object is small enough that its internal temperature is uniform at all times.

A thermometer is taken from a room where the temperature is 21°C to the outdoors, where the temperature is 5°C. After one minute, the thermometer reads 17°C. (a) What will the reading on the thermometer be after 3 more minutes? (b) When will the thermometer read 6°C?

Solve a thermometer cooling problem using Newton's Law of Cooling. Learn how to find the temperature after 3 more minutes and calculate when the thermometer will read 6°C. Explore exponential decay and cooling constant calculations, suitable for students in Grades 10-12.

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