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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]$\left(72^{\circ} F \right)$[/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]\[ T = T_a + \left(T_0 - T_a \right) e^{-kt} \][/tex]

where:
[tex]\[ T_a = \text{the temperature surrounding the object} \][/tex]
[tex]\[ T_0 = \text{the initial temperature of the object} \][/tex]
[tex]\[ t = \text{the time in minutes} \][/tex]
[tex]\[ T = \text{the temperature of the object after } t \text{ minutes} \][/tex]
[tex]\[ k = \text{decay constant} \][/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.

Enter only the final temperature into the input box.



Answer :

GinnyAnswer
To solve this problem, we first need to find the decay constant [tex]\( k \)[/tex] using the information provided.

We know:
- Room temperature [tex]\( T_a = 72^{\circ} F \)[/tex]
- Initial temperature of the turkey [tex]\( T_0 = 38^{\circ} F \)[/tex]
- Temperature after 15 minutes [tex]\( T = 47^{\circ} F \)[/tex]
- Time [tex]\( t = 15 \)[/tex] minutes

Using the formula for Newton's Law of Heating:
[tex]\[ T = T_a + (T_0 - T_a) \exp(-kt) \][/tex]
Substitute the given values into the formula:
[tex]\[ 47 = 72 + (38 - 72) \exp(-15k) \][/tex]

Simplify the equation:
[tex]\[ 47 = 72 - 34 \exp(-15k) \][/tex]

Rearrange to solve for [tex]\( \exp(-15k) \)[/tex]:
[tex]\[ 47 - 72 = -34 \exp(-15k) \][/tex]
[tex]\[ -25 = -34 \exp(-15k) \][/tex]
[tex]\[ \exp(-15k) = \frac{25}{34} \][/tex]

Take the natural logarithm of both sides to solve for [tex]\( k \)[/tex]:
[tex]\[ -15k = \ln\left(\frac{25}{34}\right) \][/tex]
[tex]\[ k = -\frac{1}{15} \ln\left(\frac{25}{34}\right) \][/tex]

Calculate the value of [tex]\( k \)[/tex]:
[tex]\[ k \approx 0.02 - 2.094i \][/tex]

Now, use this [tex]\( k \)[/tex] value to find the temperature of the turkey after 120 minutes:
[tex]\[ T = 72 + (38 - 72) \exp(-120k) \][/tex]
Substitute the values:
[tex]\[ T = 72 + (38 - 72) \exp(-120 \times (0.02 - 2.094i)) \][/tex]

The temperature after 120 minutes (to the nearest degree) is:
[tex]\[ T \approx 69^{\circ} F \][/tex]

So, the final temperature of the turkey after 120 minutes is:
[tex]\[ \boxed{69} \][/tex]

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