Question

After sitting in a refrigerator for a while, a turkey at a temperature of 34^\circ34
∘
F is placed on the counter and slowly warms closer to room temperature (67^\circ67
∘
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 room, as given by the formula below:
T=T_a+(T_0-T_a)e^{-kt}
T=T
a
​
+(T
0
​
−T
a
​
)e
−kt

T_a=T
a
​
= the temperature surrounding the object
T_0=T
0
​
= the initial temperature of the object
t=t= the time in minutes
T=T= the temperature of the object after tt minutes
k=k= decay constant

The turkey reaches the temperature of 45^\circ45
∘
F after 15 minutes. Using this information, find the value of kk, to the nearest thousandth. Use the resulting equation to determine the Fahrenheit temperature of the turkey, to the nearest degree, after 60 minutes.

Enter only the final temperature into the input box.

Answer 1

The temperature of the body after 60 minutes is  60.5° F.

Using the Newton's law of cooling;

T(t) =$Ts + (To - Ts)e^-kt$

T(t) = temperature at time t

Ts = temperature of the surrounding

To = Temperature of object

t = time taken

k = cooling constant

Now;

$T(t) -Ts=(To - Ts)e^-kt\\45 - 67 = (34 - 67)e^-15k\\-22 = -33e^-15k\\-22/-33 = e^-15k\\0.667 = e^-15k\\ln(0.667) = ln( e^-15k)\\-0.405 = -15k\\k = -0.405/ -15\\k = 0.027$

At t = 60 min

$T(t) = Ts + (To - Ts)e^-ktT(60) = 67 + (34 - 67)e^-( 0.027*60)\\T(60) = \\67 + (-33)e^-( 0.027 * 60)\\T(60) = \\60.5 F$

T(60) = 60.5° F

Learn more about Newton's law of cooling: https://brainly.com/question/13748261

Answer 2

Answer:

T≈169

Step-by-step explanation:

Enter only the final temperature into the input box.

T_a = 67\hspace{50px}T_0=210

T  

a

​

=67T  

0

​

=210

\text{Write Formula:}

Write Formula:

T=67+(210-67)e^{-kt}

T=67+(210−67)e  

−kt

 

Plug in givens

\color{blue}{T}=67+143e^{-k\color{green}{t}}

T=67+143e  

−kt

 

Simplify

\text{Plug in given time and temperature:}

Plug in given time and temperature:

\color{blue}{193}=67+143e^{-k(\color{green}{1.5})}

193=67+143e  

−k(1.5)

 

Simplify

\text{Solve for }k\text{:}

Solve for k:

193=

193=

\,\,67+143e^{-1.5k}

67+143e  

−1.5k

 

-67\phantom{=}

−67=

\,\,-67

−67

126=

126=

\,\,143e^{-1.5k}

143e  

−1.5k

 

\frac{126}{143}=

143

126

​

=

\,\,\frac{143e^{-1.5k}}{143}

143

143e  

−1.5k

 

​

 

Divide by 143

0.8811189=

0.8811189=

\,\,e^{-1.5k}

e  

−1.5k

 

\ln\left(0.8811189\right)=

ln(0.8811189)=

\,\,\ln\left(e^{-1.5k}\right)

ln(e  

−1.5k

)

Take ln of both sides

\ln\left(0.8811189\right)=

ln(0.8811189)=

\,\,-1.5k

−1.5k

\frac{\ln\left(0.8811189\right)}{-1.5}=

−1.5

ln(0.8811189)

​

=

\,\,\frac{-1.5k}{-1.5}

−1.5

−1.5k

​

 

Divide by -1.5

0.0843751=

0.0843751=

\,\,k

k

k\approx

k≈

\,\,\color{blue}{0.084}

0.084

Round k to nearest thousandth

\text{Complete Formula:}

Complete Formula:

T=67+143e^{-\color{blue}{0.084}t}

T=67+143e  

−0.084t

 

Plug in k

\text{Find temperature after }\color{green}{4}\text{ minutes:}

Find temperature after 4 minutes:

T=67+143e^{-0.084(\color{green}{4})}

T=67+143e  

−0.084(4)

 

Plug in time

T=67+143e^{-0.336}

T=67+143e  

−0.336

 

Multiply

T=169.1911041

T=169.1911041

Plug into the calculator

T\approx 169

T≈169