Question

After sitting in a refrigerator for a while, a turkey at a temperature of 36, degrees36 ∘ F is placed on the counter and slowly warms closer to room temperature (70, degrees70 ∘ 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, equals, T, start subscript, a, end subscript, plus, left bracket, T, start subscript, 0, end subscript, minus, T, start subscript, a, end subscript, right bracket, e, start superscript, minus, k, t, end superscript T=T a ​ +(T 0 ​ −T a ​ )e −kt T, start subscript, a, end subscript, equalsT a ​ = the temperature surrounding the object T, start subscript, 0, end subscript, equalsT 0 ​ = the initial temperature of the object t, equalst= the time in minutes T, equalsT= the temperature of the object after tt minutes k, equalsk= decay constant The turkey reaches the temperature of 46, degrees46 ∘ F after 30 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 115 minutes.

Answer 1

According to Newton's Law of Heating, the temperature of the turkey increases proportionally to the difference between its temperature and the room temperature. By solving the equation and using the given data, we can find the value of k and determine the temperature after a certain time.

Step-by-step explanation:

According to Newton's Law of Heating, the temperature of the turkey will increase proportionally to the difference between the temperature of the turkey and the temperature of the room. The equation is given by:

T = Ta + (T0 - Ta)e^(-kt)

Given that the turkey reaches a temperature of 46°F after 30 minutes, we can use the equation to find the value of k to the nearest thousandth. Plugging in the values:

46 = 36 + (70 - 36)e^(-30k)

Simplifying the equation:

10 = 34e^(-30k)

Taking the natural logarithm of both sides:

ln(10) = ln(34) - 30k

Solving for k:

k = (ln(34) - ln(10)) / -30

Using the resulting equation, we can determine the Fahrenheit temperature of the turkey after 115 minutes by plugging in the values:

T = 36 + (70 - 36)e^(-(ln(34) - ln(10)) / -30 * 115)

Calculating the value of T:

T = 63.194 °F