To solve this problem, we'll use Newton's Law of Cooling, which states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature. Mathematically, it's expressed as:

$\frac{dT}{dt} = -k(T(t) - T_{\text{ambient}})$

where:

• $T(t)$ is the temperature of the object at time $t$.
• $T_{\text{ambient}}$ is the ambient temperature.
• $k$ is a constant.

The solution to this differential equation is:

$T(t) = T_{\text{ambient}} + (T_0 - T_{\text{ambient}})e^{-kt}$

where:

• $T_0$ is the initial temperature of the object at $t = 0$.
• $t$ is the time elapsed.

Given:

• Initial temperature $T_0 = 185^\circ \text{F}$
• Ambient temperature $T_{\text{ambient}} = 75^\circ \text{F}$
• Temperature after 30 minutes $T(30) = 142^\circ \text{F}$

First, we need to find the constant $k$ using the given information. Then, we can use that value to find the temperature after 45 minutes.

Step 1: Find $k$

Using the formula for $T(t)$:

$142 = 75 + (185 - 75)e^{-30k}$

Simplify and solve for $k$:

$142 - 75 = 110e^{-30k}$

$67 = 110e^{-30k}$

$\frac{67}{110} = e^{-30k}$

$\ln\left(\frac{67}{110}\right) = -30k$

$k = -\frac{1}{30} \ln\left(\frac{67}{110}\right)$

Step 2: Find the temperature after 45 minutes

Now, using the value of $k$ we calculated, we substitute it into the formula to find $T(45)$:

$T(45) = 75 + 110e^{-45k}$

Let's calculate this step by step.

Calculation

I'll compute the value of $k$ and then find $T(45)$.import math

Given values

T_ambient = 75 # Ambient temperature T0 = 185 # Initial temperature of the turkey T_30 = 142 # Temperature after 30 minutes t1 = 30 # Time in minutes (half an hour) t2 = 45 # Time in minutes (45 minutes)

Calculate the constant k

k = -(1 / t1) * math.log((T_30 - T_ambient) / (T0 - T_ambient))

Calculate the temperature after 45 minutes

T_45 = T_ambient + (T0 - T_ambient) * math.exp(-k * t2) T_45The temperature of the turkey after 45 minutes is approximately 127.29°F.

Would you like more details or have any other questions?

Here are some related questions you might find interesting:

1. How would the temperature change after 1 hour?
2. What if the ambient temperature was different?
3. How does the rate constant $k$ affect the cooling process?
4. What happens if the turkey was removed from the oven at a different initial temperature?
5. How could this model be adjusted for different cooling conditions?

Tip: Newton's Law of Cooling can be applied to various scenarios, such as cooling beverages or even forensic investigations to estimate the time of death.