https://file.aiquickdraw.com/mathbot/file/1729377024004zsjxq4c1.jpg

This problem involves Newton's Law of Cooling, which describes the temperature of an object cooling down in a surrounding environment.

### Newton's Law of Cooling formula:
\[
T(t) = T_s + (T_0 - T_s) e^{-kt}
\]
Where:
- \( T(t) \) is the temperature of the object at time \( t \).
- \( T_s \) is the surrounding temperature.
- \( T_0 \) is the initial temperature of the object.
- \( k \) is a constant.
- \( t \) is the time elapsed.

### Given Data:
- Initial temperature of the turkey, \( T_0 = 185^\circ F \).
- Surrounding temperature, \( T_s = 75^\circ F \).
- After 30 minutes (0.5 hours), the temperature of the turkey is \( 150^\circ F \).

### Part (a):
We need to find the temperature of the turkey after 45 minutes (or 0.75 hours).

1. **Step 1: Find the constant \( k \)**  
   Use the data given for 30 minutes to find \( k \). The temperature at 0.5 hours is \( 150^\circ F \), so we substitute into the cooling formula:
   \[
   150 = 75 + (185 - 75) e^{-k(0.5)}
   \]
   Simplifying:
   \[
   150 = 75 + 110 e^{-0.5k}
   \]
   \[
   75 = 110 e^{-0.5k}
   \]
   \[
   \frac{75}{110} = e^{-0.5k}
   \]
   \[
   \ln\left(\frac{75}{110}\right) = -0.5k
   \]
   Solving for \( k \):
   \[
   k = -2 \ln\left(\frac{75}{110}\right)
   \]
   Compute \( k \).

2. **Step 2: Find the temperature at 45 minutes**  
   Now that we have \( k \), substitute \( t = 0.75 \) hours into the Newton's Law of Cooling equation to find the temperature at 45 minutes.

### Part (b):
We need to find when the turkey will cool to \( 100^\circ F \).

1. **Step 1: Use the cooling formula**  
   Set \( T(t) = 100 \), and solve for \( t \):
   \[
   100 = 75 + (185 - 75) e^{-kt}
   \]
   Simplifying:
   \[
   100 = 75 + 110 e^{-kt}
   \]
   \[
   25 = 110 e^{-kt}
   \]
   \[
   \frac{25}{110} = e^{-kt}
   \]
   \[
   \ln\left(\frac{25}{110}\right) = -kt
   \]
   Solving for \( t \):
   \[
   t = \frac{\ln\left(\frac{25}{110}\right)}{-k}
   \]
   Compute \( t \).

I will now compute the constants and the values needed to solve both parts.Here are the results:

1. The cooling constant \( k \) is approximately **0.766**.
2. After 45 minutes (0.75 hours), the temperature of the turkey will be approximately **136.93°F**.
3. The turkey will cool to **100°F** after about **1.93 hours** (approximately 1 hour and 56 minutes).

Would you like any further details or clarifications?

Here are five related questions to explore:
1. How would the result change if the room temperature were different?
2. Can this same cooling model be used for other materials? Why or why not?
3. What assumptions are made in Newton's Law of Cooling?
4. How would the cooling rate differ if the turkey started at a lower temperature, such as 150°F?
5. How does the value of \( k \) affect the cooling process?

**Tip:** Newton's Law of Cooling is applicable to objects cooling down or heating up toward a stable environment temperature, but it assumes uniform heat transfer without external interference.

A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F. (a) If the temperature of the turkey is 150°F after half an hour, what is the temperature after 45 minutes? (b) When will the turkey have cooled to 100°F?

This problem involves calculating the cooling rate of a roast turkey using Newton's Law of Cooling. Learn how to apply the exponential decay formula to find the turkey's temperature after 45 minutes and when it will cool to 100°F, given the surrounding temperature is 75°F. Suitable for Grades 11-12 or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation