Question 1178344
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°?
<pre>Formula for Newton's Law of cooling: {{{matrix(1,3, T(t), "=", T[s] + (T[o]  -  T[s])e ^(- kt))}}} where: {{{t}}} is the time at a COOLED temperature 
                                                                        {{{T(t)}}} is the TEMPERATURE (T) at a given time (t) 
                                                                        {{{T[s]}}} is the SURROUNDING temperature 
                                                                        {{{T[o]}}} is the ORIGINAL/INITIAL temperature 
                                                                        {{{ k}}} is the CONSTANT or COOLING rate</pre><pre>

{{{matrix(1,3, T(t), "=", T[s] + (T[o]  -  T[s])e ^(- kt))}}} then becomes: {{{matrix(1,3, T(30), "=", 75 + (185 - 75)e^(- 30k))}}} -- Substituting 30 for t, 75 for {{{T[s]}}}, and 185 for {{{T[o]}}}
                                           {{{matrix(1,3, 150, "=", 75 + (185 - 75)e^(- 30k))}}} -- Substituting 150 for {{{T(30)}}}
                                           {{{matrix(4,3, 150 - 75, "=", 110e^(- 30k), 75, "=", 110e^(- 30k), 75/110, "=", e^(- 30k), 15/22, "=", e^(- 30k))}}} 
                                           {{{matrix(1,3, - 30k, "=", ln(15/22))}}} ----- Converting to LOGARITHMIC (Natural) form 

                    CONSTANT/COOLING RATE, or {{{highlight_green(matrix(1,5, k, "=", ln(15/22)/- 30, "=", highlight(matrix(1,4, - 0.012766408^o, "F,", per, minute))))}}}

{{{matrix(1,3, T(t), "=", T[s] + (T[o]  -  T[s])e ^(- kt))}}} becomes: {{{matrix(1,3, T(50), "=", 75 + (185 - 75)e^(- .012766408(50)))}}} -- Substituting 75, 185, - .012766408, and 50  
                                                                            for {{{T[s]}}}, {{{T[o]}}}, k, and t, respectively
                                     {{{matrix(1,3, T(50), "=", 75 + 110e^(- .012766408(50)))}}}
   Temperature after 50 minutes, or {{{highlight_green(matrix(1,3, T(50), "=", highlight(matrix(1,4, 133.0996678, "=", 133^o, F))))}}}

{{{matrix(1,3, T(t), "=", T[s] + (T[o]  -  T[s])e ^(- kt))}}} becomes: {{{matrix(1,3, 105, "=", 75 + (185 - 75)e^(- .012766408t)))}}} -- Substituting 105, 75, 185, and - .012766408,                                   
                                                                           for {{{T(t)}}}, {{{T[s]}}}, {{{T[o]}}}, and k, respectively
                                      {{{matrix(1,3, 105, "=", 75 + 110e^(- .012766408t))}}} 
                                      {{{matrix(4,3, 105 - 75, "=", 110e^(- .012766408t), 30, "=", 110e^(- .012766408t), 30/110, "=", e^(- .012766408t), 3/11, "=", e^(- .012766408t))}}} 
                                      {{{matrix(1,3, - .012766408t, "=", ln(3/11))}}} ----- Converting to LOGARITHMIC (Natural) form 

TIME taken for turkey to cool to 105<sup>o</sup>, or {{{highlight_green(matrix(1,5, t, "=", ln(3/11)/(- .012766408), "=", highlight(matrix(1,4, 101.7735752, "=", 102, minutes))))}}}</pre>