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.