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After sitting in a refrigerator for a while, a turkey at a temperature of [tex]38^{\circ} F[/tex] is placed on the counter and slowly warms closer to room temperature [tex]72^{\circ} F[/tex]. Newton's Law of Heating explains that the temperature of the turkey will increase proportionally to the difference between the temperature of the turkey and the temperature of the room, as given by the formula below:

[tex]\[
\begin{array}{l}
T = T_a + \left(T_0 - T_a\right) e^{-k t} \\
T_a = \text{the temperature surrounding the object} \\
T_0 = \text{the initial temperature of the object} \\
t = \text{the time in minutes} \\
T = \text{the temperature of the object after } t \text{ minutes} \\
k = \text{decay constant}
\end{array}
\][/tex]

The turkey reaches the temperature of [tex]47^{\circ} F[/tex] after 15 minutes. Using this information, find the value of [tex]k[/tex], to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the nearest degree, after 120 minutes.



Answer :

GinnyAnswer
Certainly! To solve this problem, we will use Newton's Law of Heating, which is given by:

[tex]\[ T = T_a + (T_0 - T_a) e^{-kt} \][/tex]

where
- [tex]\( T \)[/tex] is the temperature of the object after time [tex]\( t \)[/tex]
- [tex]\( T_a \)[/tex] is the surrounding temperature
- [tex]\( T_0 \)[/tex] is the initial temperature of the object
- [tex]\( k \)[/tex] is the decay constant
- [tex]\( t \)[/tex] is the time in minutes

We are given the following values:
- [tex]\( T_0 = 38^\circ F \)[/tex]
- [tex]\( T_a = 72^\circ F \)[/tex]
- After 15 minutes ([tex]\( t = 15 \)[/tex]), the temperature of the turkey is [tex]\( 47^\circ F \)[/tex]

### Step 1: Determining the decay constant [tex]\( k \)[/tex]
To find [tex]\( k \)[/tex], we first substitute the known values into the formula and solve for [tex]\( k \)[/tex].

[tex]\[ 47 = 72 + (38 - 72) e^{-k \cdot 15} \][/tex]

Simplify the equation:

[tex]\[ 47 = 72 - 34 e^{-15k} \][/tex]

Rearrange to isolate the term with [tex]\( e \)[/tex]:

[tex]\[ 47 - 72 = -34 e^{-15k} \][/tex]

[tex]\[ -25 = -34 e^{-15k} \][/tex]

[tex]\[ \frac{25}{34} = e^{-15k} \][/tex]

Take the natural logarithm of both sides to solve for [tex]\( k \)[/tex]:

[tex]\[ \ln\left(\frac{25}{34}\right) = -15k \][/tex]

[tex]\[ k = -\frac{\ln\left(\frac{25}{34}\right)}{15} \][/tex]

Calculating the numerical value gives us approximately:

[tex]\[ k \approx 0.02 \][/tex]

### Step 2: Determining the temperature after 120 minutes

Now we use the decay constant [tex]\( k \approx 0.02 \)[/tex] to find the temperature [tex]\( T \)[/tex] after 120 minutes.

[tex]\[ T = T_a + (T_0 - T_a) e^{-kt} \][/tex]

Substitute the values into the formula:

[tex]\[ T = 72 + (38 - 72) e^{-0.02 \cdot 120} \][/tex]

Simplify the expression inside the exponential function:

[tex]\[ T = 72 + (38 - 72) e^{-2.4} \][/tex]

Calculate the exponential term:

[tex]\[ e^{-2.4} \approx 0.0907 \][/tex]

Now compute the temperature:

[tex]\[ T \approx 72 + (38 - 72) \cdot 0.0907 \][/tex]

[tex]\[ T \approx 72 + (-34) \cdot 0.0907 \][/tex]

[tex]\[ T \approx 72 - 3.08 \][/tex]

[tex]\[ T \approx 69^\circ F \][/tex]

### Final Answer
1. The decay constant [tex]\( k \)[/tex] is approximately [tex]\( 0.02 \)[/tex] to the nearest thousandth.
2. The temperature of the turkey after 120 minutes is approximately [tex]\( 69^\circ F \)[/tex] to the nearest degree.

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