Question 1178344
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A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room 
where the temperature is 75°F. (Round your answers to the nearest whole number.)
(a) If the temperature of the turkey is 150°F after half an hour, what is the temperature after 50 minutes?
T(50) =

(b) When will the turkey have cooled to 105°?
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<pre>
The Newton law of cooling states that the temperature of the roast turkey in the room is this function of time t


    T(t) = {{{75 + (185-75)*e^(-kt)}}} = {{{75 + 110*e^(-kt)}}}.


where "k" is the decay constant.  

At t= 30 minutes  T(t)= 150°F,  which gives you an equation to find the decay constant k:


    150 = 75 + 110*e^(-k*30)

    110*e^(-k*30) = 150 - 75 = 75

    e^(-30k) = {{{75/110}}} = 0.6818

    - 30k = ln(0.6818)

    k = {{{-ln((0.6818))/30}}} = 0.01277.


Thus the decay constant k is found.


        Now I am in position to answer questions (a) and (b).


The temperature after 50 minutes is


    T(50) = 75 + 110*e^(-0.01277*50) = 75 + 110*2.71828^(-0.6385) = 133°F.      <U>ANSWER to question (a)</U>



To find the time getting 105°F, use and solve this equation 


    105 = {{{75 + 110*e(-0.01277t)}}}


    e(-0.01277*t) = {{{(105-75)/110}}} = 0.2727

    -0.01277*t = ln(0.2727)

    t = {{{-ln(0.2727)/0.01277}}} = 102 minutes   (rounded).       <U>ANSWER to question (b)</U>
</pre>

Solved.