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After sitting out of a refrigerator for a while, a turkey at room temperature [tex]\(\left(70^{\circ} F \right)\)[/tex] is placed into an oven. The oven temperature is [tex]\(310^{\circ} F\)[/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 oven, as given by the formula below:

[tex]\[ T = T_a + \left(T_0 - T_a \right) e^{-k t} \][/tex]

Where:
- [tex]\( T_a \)[/tex] = the temperature surrounding the object
- [tex]\( T_0 \)[/tex] = the initial temperature of the object
- [tex]\( t \)[/tex] = the time in hours
- [tex]\( T \)[/tex] = the temperature of the object after [tex]\( t \)[/tex] hours
- [tex]\( k \)[/tex] = decay constant

The turkey reaches the temperature of [tex]\(122^{\circ} F\)[/tex] after 2 hours. 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 5.5 hours.

Enter only the final temperature into the input box.



Answer :

GinnyAnswer
To solve this problem, we utilize Newton's Law of Heating, given by the formula:

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

Let's break down the solution step-by-step:

1. Identify Given Values:
- [tex]\( T_a = 310^\circ \text{F} \)[/tex] (oven temperature)
- [tex]\( T_0 = 70^\circ \text{F} \)[/tex] (initial temperature of the turkey)
- [tex]\( T = 122^\circ \text{F} \)[/tex] (temperature of the turkey after 2 hours)
- [tex]\( t = 2 \)[/tex] hours (time when the temperature is measured)

2. Set Up the Equation with the Given Values:
Substitute the known values into the formula:
[tex]\[ 122 = 310 + (70 - 310) \cdot e^{-2k} \][/tex]

3. Simplify the Equation:
[tex]\[ 122 = 310 - 240 \cdot e^{-2k} \][/tex]
[tex]\[ 122 - 310 = -240 \cdot e^{-2k} \][/tex]
[tex]\[ -188 = -240 \cdot e^{-2k} \][/tex]
[tex]\[ \frac{188}{240} = e^{-2k} \][/tex]

4. Solve for [tex]\( k \)[/tex]:
Take the natural logarithm on both sides to solve for [tex]\( k \)[/tex]:
[tex]\[ \ln \left(\frac{188}{240}\right) = -2k \][/tex]
[tex]\[ k = -\frac{1}{2} \ln \left(\frac{188}{240}\right) \][/tex]
[tex]\[ k \approx 0.122 \][/tex]

Thus, the decay constant [tex]\( k \)[/tex] is approximately [tex]\( 0.122 \)[/tex] to the nearest thousandth.

5. Use the Decay Constant to Find the Temperature After 5.5 Hours:
We use the same formula with [tex]\( t = 5.5 \)[/tex]:
[tex]\[ T = 310 + (70 - 310) \cdot e^{-0.122 \cdot 5.5} \][/tex]
Calculate the exponent:
[tex]\[ e^{-0.122 \cdot 5.5} \approx 0.397 \][/tex]
Substitute back into the equation:
[tex]\[ T = 310 + (70 - 310) \cdot 0.397 \][/tex]
[tex]\[ T = 310 - 240 \cdot 0.397 \][/tex]
[tex]\[ T \approx 310 - 95.92 \][/tex]
[tex]\[ T \approx 214.08 \][/tex]

6. Round to the Nearest Degree:
The temperature of the turkey after 5.5 hours is approximately [tex]\( 214^\circ \text{F} \)[/tex].

Therefore, after 5.5 hours, the temperature of the turkey is [tex]\( 187^\circ \text{F} \)[/tex].

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