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$Question Show Examples After sitting out of a refrigerator for a while, a turkey at room temperature (70^(@)F) is placed into an oven. The oven temperature is 325^(@) F. Newton's Law of Heating explains that the temperature of the turkey will increase proportionally to the difference between the temperature of the turkey and the temperature of the oven, as given by the formula below: {:[qquad T=T_(a)+(T_(0)-T_(a))e^(-kt)],[T_(a)=" the temperature surrounding the object "],[T_(0)=" the initial temperature of the object "],[t=" the time in hours "],[T=" the temperature of the object after "t" hours "],[k=" decay constant "]:} The turkey reaches the temperature of 116^(@)F after 2-5 hours. Using this information, find the value of k, to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the nearest degree, after 5.5 hours. Enter only the final temperature into the input box. Answer Attemps z out of 2 T=57 Submit Answert$

Question $\newline$ Show Examples $\newline$ After sitting out of a refrigerator for a while, a turkey at room temperature $\left(70^{\circ} \mathrm{F}\right)$ is placed into an oven. The oven temperature is $325^{\circ}$ F. 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: $\newline$ $\begin{array}{l} \qquad T=T_{a}+\left(T_{0}-T_{a}\right) e^{-k t} \\ T_{a}=\text { the temperature surrounding the object } \\ T_{0}=\text { the initial temperature of the object } \\ t=\text { the time in hours } \\ T=\text { the temperature of the object after } t \text { hours } \\ k=\text { decay constant } \end{array}$ $\newline$ The turkey reaches the temperature of $116^{\circ} \mathrm{F}$ after $2-5$ hours. Using this information, find the value of $k$ , to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the nearest degree, after $5$ . $5$ hours. $\newline$ Enter only the final temperature into the input box. $\newline$ Answer Attemps z out of $2$ $\newline$ $T=57$ $\newline$ Submit Answert

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Solve linear equations with variables on both sides: word problems

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Q. Question

\newline

Show Examples

\newline

After sitting out of a refrigerator for a while, a turkey at room temperature

\left(70^{\circ} \mathrm{F}\right)

is placed into an oven. The oven temperature is

325^{\circ}

F. 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:

\newline

\begin{array}{l} \qquad 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 hours } \\ T=\text { the temperature of the object after } t \text { hours } \\ k=\text { decay constant } \end{array}

\newline

The turkey reaches the temperature of

116^{\circ} \mathrm{F}

after

2-5

hours. Using this information, find the value of

k

, to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the nearest degree, after

5

5

hours.

\newline

Enter only the final temperature into the input box.

\newline

Answer Attemps z out of

2

\newline

T=57

\newline

Submit Answert

Plug in values and find k: First, let's plug in the values we know into the formula to find k. We have $T_a = 325$ , $T_0 = 70$ , $T = 116$ , and $t = 2.5$ . So, $116 = 325 + (70 - 325)e^{(-k*2.5)}$ .
Simplify the equation: Now, let's simplify the equation. We get $116 = 325 - 255e^{-k \times 2.5}$ .
Isolate the exponential term: Subtract $325$ from both sides to isolate the exponential term. We get $-209 = -255e^{-k \cdot 2.5}$ .
Solve for $e^{(-k*2.5)}$ : Divide both sides by $-255$ to solve for $e^{(-k*2.5)}$ . We get $e^{(-k*2.5)} = \frac{209}{255}$ .
Take natural logarithm: Now, take the natural logarithm of both sides to solve for $-k \times 2.5$ . We get $-k \times 2.5 = \ln(\frac{209}{255})$ .
Solve for $k$ : Divide both sides by $-2.5$ to solve for $k$ . We get $k = -\ln(\frac{209}{255})/2.5$ .
Calculate the value of k: Calculate the value of k. $k \approx -\ln(\frac{209}{255})/2.5 \approx 0.231$ .